Dear Calvin, if any portion of the world is described by probabilistic wave functions, then the whole world has to be. It's easy to show it. Take a decaying nucleus, connect it to a hammer that kills a cat a that also makes the Sun explode into 2 pieces.
The nucleus is evolving into a linear superposition of "decayed" and "not yet decayed" states. Correspondingly, because of the mechanism, the Sun has to evolve into a linear superposition of "exploded" and "not yet exploded". These two states have different gravitational fields. It proves that in general, the evolution produces linear superpositions of states with different gravitational fields, so the gravitational fields - and any other physical properties of the world - have to be described by linear operators just like any other physical property.
If some building blocks can only be predicted probabilistically, it's clear that everything else that may be affected by these building blocks can only be predicted probabilistically, too. The whole world, including gravitational fields, interacts with itself, so clearly the gravitational waves have to be ultimately described by quantum physics, too.
Gravitons are physical particles that are quanta of gravitational waves. They have to exist because
- gravitational waves exist
- the energy stored in a frequency $f$ classical state is always a multiple of $E=hf$
The first point was indirectly, but pretty much conclusively, proved by the observation of the binary pulsar that changes its frequency in the right way, as it emits the gravitational waves and loses energy. Everything agrees with general relativity beautifully. The 1993 physics Nobel prize was given for this confirmation of the gravitational waves.
The second point is a trivial consequence of Schrödinger's equation. Take a classical gravitational wave of frequency $f$ - e.g. similar to what is emitted by the binary pulsar. Divide the corresponding quantum state to energy eigenstate components. They go like
If all classical observables evaluated in this state are periodic with periodicity $1/f$, it's trivial to see that all the energy differences $E_m-E_n$ must be multiples of $E_0=hf$, so the energy can only be added to the gravitational waves by quanta, the gravitons.
One may also derive the gravitons and their polarizations from the linear approximation of the quantized general relativity. While general relativity has problems at higher-loop level - strongly quantum effects that affect spacetime - this approximation has to work very well, is consistent, and implies the existence of gravitons, spin 2 massless particles, too.
In quantum gravity, virtual gravitons are the messengers of the gravitational force in the same way as photons are messengers of the electromagnetic force. Gravitons and photons have a different spin but otherwise the analogy between them is - and must be - much tighter than the wording of your question is willing to admit. They're described as quantum fields and the effective quantum field theory with Feynman diagrams etc. has to work, with the same interpretation, at least in some approximation.
Also, it's not true that the non-gravitational forces can't be geometric. The Kaluza-Klein theory explains the electromagnetic field as "twists" that include an extra dimension of space. Whether or not the extra dimensions of space have this simple form, string theory generalizes the Kaluza-Klein lesson and every field and particle species may be viewed as a component of a generalized geometry field - because the geometry is generalized in such a way that it includes all other fields as well. Some of the string scenarios are very close to the Kaluza-Klein theory, some of them are far from it, but all of them confirm that gravity ultimately comes from the same underlying physics as everything else.
The term "string theory" may sound too narrow-minded in this context. There are very many different ways how to describe its physics, including gauge theory (just an old quantum field theory!) that lives on the holographic boundary of the AdS space. All of them agree that gravitons have to exist and they're just one particle species among many that share a common origin. So the term "string theory" in all these discussions really means "everything we have ever learned about quantum gravity that has worked".