Is it possible for a particle to decay via gravity? Is it possible for a particle to decay via gravity? I know gravity is immensely weaker than the other forces, but all the other forces interact with particles. Do we need an understanding of quantum gravity to know if this is possible? Are there any theories where this is possible?
 A: The closest gravitation decay process we understand is Hawking radiation. A black holes with area $A=8\pi M^2$ is composed of $N$ Planck masses or with $M=Nm_{pl}$. The emission of a quanta or a boson, so $N\rightarrow N-1$ is a sort of decay process, and will repeat until the black hole is evaporated.
A: It is impossible in perturbative QFT.
The quantum field theory of gravity is not renormalizeable, but you can always calculate processes in perturbation theory in the effective field theory at low energy. In this effective field theory, matter couples to gravity with interaction Lagrangian
$$\mathcal{L}_{\rm int} \sim \frac{1}{\sqrt{32\pi G}} T_{\mu\nu} h^{\mu\nu}.$$
Naively, we would like to have a "beta-like" decay via the diagram:

But because the stress tensor $T$ only couples particles to themselves, the particle $\psi'$ is actually the same as the particle $\psi$ and this is not a decay at all. In beta decay, the case is different, as the $W$ carries weak charge, so e.g. neutrons can decay to protons.
This means that the decay is impossible in the effective theory of gravity. On the other hand, a decay through Hawking radiation, as described by Anders Sandberg above and others, is in principle possible.
A: The closest I have seen is decay through virtual black holes. 
This paper, for example, works out proton decay rates due to black holes if there are extra large dimensions. The paper starts by heuristically estimating the lifetime in normal space-time, estimating the lifetime as $\tau \sim \frac{1}{m_p}\left(\frac{M_{Pl}}{m_p}\right)^4 \sim 10^{45}$ years. Since we do not have a proper quantum field theory doing the full decay calculation is not possible yet; how much to trust the heuristic argument remains to be seen. There are plenty of potential complications. 
A: 
Is it possible for a particle to decay via gravity?

This is not the case in the Standard Model, which does not include gravity.
This is also not the case in General Relativity, which doesn't have a particle based mechanism for particle decay, although General Relativity does recognize that the relativistic mass of a system can change in a conversion of gravitational energy to other kinds of matter-energy, or visa versa, under certain circumstances at a macroscopic level.
It is conceivable that particle decay via gravity could happen in a realistic theory of quantum gravity. But, certainly, there is no credible experimental evidence of such a decay being observed in the real world at this time. Since that would possibly be very hard to observe, however, this isn't necessarily remarkable and doesn't necessarily imply strongly that it isn't possible.

I know gravity is immensely weaker than the other forces, but all the
other forces interact with particles. Do we need an understanding of
quantum gravity to know if this is possible?

Yes. We do need an understanding of quantum gravity to know if this is possible, since it is not otherwise possible, and understanding an inherently quantum gravitational concept requires an understanding of quantum gravity.
While there are numerous proposals for quantum gravity theories, none of them have achieved consensus support or even really credible claims that they are likely to be a correct theory of quantum gravity. So, the short answer is that we don't know and probably aren't very close to knowing.
This said, there are many qualitative assumptions that are typically made about broad general classes of quantum gravity theories that can help us to make educated guesses about what should and should not be possible by analogy to what we know about the Standard Model, about General Relativity, and about proposed quantum gravity theories.
Presumably, any decay via gravity in a quantum gravity would, like decays via other forces, have to preserve all conserved quantum numbers and conserved quantities such as electromagnetic charge, color charge,  mass-energy, CPT, baryon number, lepton number, and linear and angular momentum. There is also good reason to think that gravitational decays would not violate CP symmetry.
But, it is also the case that one or more of these quantities which are conserved in the Standard Model might not be conserved in a quantum gravity theory, with mass-energy conservation, for example, being doubtful as a globally conserved quantity in a quantum gravity theory.

Are there any theories where this is possible?

Yes. There are theories where this is possible. A recent pre-print addressing this possibility states:

In extended models of gravity a non-minimal coupling to matter has been
assumed to lead to irreversible particle creation. In this paper we
challenge this assumption. We argue that a non-minimal coupling of the
matter and gravitational sectors results in a change in
particle-momentum on a cosmological timescale, irrespective of
particle creation or decay. We further argue that particle creation
or decay associated with a non-minimal coupling to gravity could only
happen as a result of significant deviations from a homogeneous
Friedmann-Robertson-Walker geometry on microscopic scales, and provide
a phenomenological description of the impact of particle creation or
decay on the cosmological evolution of the density of the matter
fields.

R.P.L. Azevedo, P.P. Avelino, "Particle creation and decay in nonminimally coupled models of gravity" (Submitted on 18 Jan 2019 (v1), last revised 29 Mar 2019 (this version, v2)) (minor punctuation and spelling corrections made editorially to abstract language quoted above without indication).
This paper concludes as follows:

In this work we challenged the assumption that the NMC between
geometry and the matter fields might be responsible for particle
creation/decay in the absence of significant perturbations to the FLRW
metric on microscopic scales. We have argued that there is only one
consistent interpretation for the modification to the evolution of the
energy density of a fluid made of soliton-like particles associated to
the the NMC between the gravitational and the matter fields in a FLRW
universe: a change in particle-momentum on a cosmological timescale
(rather than particle creation or decay). We have considered the
possibility that perturbations to the FLRW geometry on microscopic
scales, eventually in association to significant extensions to the NMC
theory of gravity studied in the present paper, may be responsible for
particle creation or decay. We have also have provided a
phenomenological description of particle creation/decay by defining an
“effective Lagrangian” which incorporates these effects.

UPDATE September 30, 2020:

Is it possible for a particle to decay via gravity?

One nuance of your question (possibly unintended) that I wasn't considering when I wrote this answer is that your question literally is asking a a particle can decay via gravity, and not if particles can decay via gravity.
As I noted in the original answer:

Presumably, any decay via gravity in a quantum gravity would, like
decays via other forces, have to preserve all conserved quantum
numbers and conserved quantities such as electromagnetic charge, color
charge, mass-energy, CPT, baryon number, lepton number, and linear and
angular momentum.

At a minimum, these conservation laws imply that many fundamental standard model particles gravitational decays are ruled out, although I haven't done the analysis to confirm that all single fundamental particle gravitational decays are definitively ruled out.
We also know, because the Standard Model, ignoring entirely the possibility of any gravitational decays of fundamental particles, predicts the observed decays of fundamental particles to high precision, that the deviation from the Standard Model background caused by gravitational decays of fundamental particles, if they happen at all, give rise to a quantitatively quite small effect. This is what we would expect since the strength of the force of gravity (in Newtonian approximation) at particle physics scale distances and masses is negligible relative to the strength of the three Standard Model forces.
To illustrate the idea that conservation laws limit the possible types of gravitational decays of single fundamental particles, we need to explore two cases in a preliminary analysis.
Case One
In one case, the end result of the decay is purely gravitons, in a close analog to a matter-antimatter annihilation to a a pair of photons.
In the photon case, the conservation of conserved quantities implies that a matter-antimatter annihilation always produces a pair of photons and not just a single photon.
In the analogous gravitational decay case, the conservation of conserved quantities implies that a matter-antimatter annihilation always produces a pair of gravitons and not just a single graviton.
So, there is no way that a single fundamental particle could decay into a single graviton.
Case Two
In case two, a particle emits a graviton and also something else which is different that the original particle and, as a result of conservation of mass-energy, involves a final state of the decay with a particle of lower rest mass than the original particle.
Since a hypothetical graviton would not carry baryon number, lepton number, color charge, or electromagnetic charge, all of those conserved quantities would have to end up in decay products other than the graviton produced by the decay.
So, for example, it would be impossible (with mainstream minimal assumptions) for an up quark, down quark, or electron, or electron neutrino, to decay via a single graviton, because there is no other particle or combination of particles, to which the up quark or electron could decay with less rest mass that would preserve, (1) in the case of the up or down quark quark, baryon number and electromagnetic charge and color charge, (2) in the case of the electron, lepton number and electromagnetic charge, and (3) in the case of the electron-neutrino, lepton number.
So, there is no way that a single first generation fundamental fermion could decay gravitationally.
Likewise, because gluons have color charge, there is no way that a single gluon could decay gravitationally to one or more gravitons (and/or one or more photons and and/or one or more fundamental leptons). This is because all of these decay products lack color charge and color charge is conserved. Thus, any gravitational decay of a quark or gluon would have to have quarks or gluons in addition to gravitons a decay products.
While rigorous proof is more involved with second and third generation fundamental fermions, W bosons, Z bosons, Higgs bosons and photons, most or all of these particles, by themselves, could also not decay to another particle solely by emission of a single graviton.
In all or most of these cases, conservation laws would demand that any decay involving a graviton also include additional particles, and that there be more than one particle in the pre-decay state. See, e.g., here (assuming that gravitational decays would involve more than one source particle).
For example, conservation of angular momentum will generally prohibit the decay of a single spin-1/2 fundamental fermion into a single spin-2 graviton and a different single spin-1/2 fundamental fermion.
This paper, in an analysis of possible gravitational decays, suggests that scenarios involving Higgs bosons, W and Z bosons, and top quarks are likely to be most significant in a quantum gravity model involving large extra-dimensions in which gravity can propagate, but the other Standard Model forces are confined to a brane with 3+1 dimensions (a common assumption in string theory). See also this paper from the year 2000, twelve years before the Higgs boson was experimentally discovered, discussing gravitational decays of Higgs bosons under different assumptions.
A: This is not an answer, but more of an observation. It is worth noting that most of the heavier particles decayed as a result of the expansion of the universe, a gravitational effect. The expansion is accompanied by gravitational redshift, causing particles to eventually not have enough momentum to collide and form the heavier particles. When this happens, the equilibrium distributions of these heavier particles fall out of equilibrium, as their decay rate now exceeds the production rate. Although the decay itself is not mediated by gravitons, gravity certainly plays a role in orchestrating it.
A different scenario is where a charged particle circles a black hole, and emits synchrotron radiation as a result. While conserved quantum numbers remain unaffected, a particle does decay into a less excited state, accompanied by the emission of photons. The point is that turning on a background field does affect the dynamics of a quantum field theory.
A: There are some great answers above; however, I'm going to take a different track only because it hasn't been brought up:
Firstly: By particle I'm guessing you mean some elementary object with mass, spin, charge et cetera. Or perhaps a composite of such objects that decays to a lower energy state.
And you're asking if such an object can decay via gravitational energy (ie. releasing or absorbing gravitational energy)
If this is what you meant then the answer is YES!!! General Relativity does this all by itself!
Consider two black holes of essentially arbitrary mass, spin, and charge in binary orbit. General relativity predicts that the system slowly decays via the release of gravitational wave energy.
When the two finally merge an enormous amount of energy is released in the form of gravitational waves as LIGO has in fact proven (and been awarded the 2017 Nobel prize for).
ALSO:  If you wish to confine yourself to the standard model particles of quantum theory, the answer is still YES. The very fact that the particles making up a star can decay into a neutron star and sometimes a black hole is also proof of such a concept. Again, simply using General relativity.
I am amazed that quantum Theory was in everyone's answers when the question asks specifically if we:

need an understanding of quantum gravity to know if this is possible?

A: My message corresponds to my thoughts on this issue and therefore is not described in the literature. The law of interaction between masses and charges has the form
$$\vec F=\frac{(ie+m1\sqrt{G})(ie+m2\sqrt{G})}{r^3}\vec r$$
 But in nature there are either large masses and a small charge, or small masses of elementary particles and the charge is large compared to the mass, so the interaction between the imaginary charge and the actual mass is not described. The contribution of the imaginary part of the force is determined by the formula
$$\vec F=Re\vec F+Im\vec F sin\omega t$$
 The frequency must be taken Compton. Why am I saying this, the fact is that imaginary charges form the imaginary part of the mass, and the decay of an elementary particle forms. Real charges - masses cannot have this effect.
