I understand that inertial mass, at least in part, comes from the inertia of energy in the zero momentum frame or rest frame of some physical system. So for a static charge the corresponding field energy is positive giving rise to a positive electromagnetic inertial mass. Is it also possible to define an inertial mass for the gravitational field like the above and if so, is it positive or negative?
There is no simple answer to this question. General relativity does not generically have any way to define the total energy of a system; it only has conserved, scalar measures of the total energy of a system in particular special cases, such as asymptotically flat spacetimes. Even then, there is more than one such measure of energy. For instance, we have the ADM energy and Bondi energy, both of which are conserved for an asymptotically flat spacetime. They differ because the Bondi energy doesn't include energy being radiated away to infinity by gravitational waves, whereas the ADM energy does.
Note also that these energies are not additive as in Newtonian mechanics.
The basic reason for all of this complication is that the equivalence principle says that the gravitational "field" in the Newtonian sense is unobservable. A free-falling observer (which is the preferred kind of inertial observer in GR) always measures the gravitational field to be zero. Therefore we can't have, as we do in Newtonian mechanics, an equation that defines the energy density in terms of the gravitational field. When relativists talk about the "gravitational field," they really mean the curvature, not the meaningless Newtonian $g$.
There seems to have been a lot of dispute and confusion in comments and answers about whether the energy in the gravitational field is positive or negative. Well, in general, it's not even defined, but in the cases where it's defined, it can be either positive or negative. For example, a Schwarzschild black hole has a positive ADM energy, which is equal to its inertial mass, and this energy is entirely due to its gravitational field (since it's a vacuum solution to the Einstein field equations, i.e., no matter is present). On the other hand, the gravitational energy of the earth is certainly negative, because the earth is nonrelativistic, and in the nonrelativistic approximation energy is additive and gravitational energies are negative. Gravitational waves have positive energy.
A partial answer. It's possible to build up very simplified models of static self-gravitating "stars" where it's manifest that the total mass of the star is less than the volume integral of the rest density of the matter the star is made of. And that the difference is just the gravitational potential energy of the star.
Inertial mass of all known physical bodies is positive. For ordinary bodies, the gravitational energy contribution (which is negative) must be therefore smaller than other contributions. Try to calculate gravitational self-energy of 1kg package of sugar. It is minuscule compared to rest energy $mc^2$.
$\begingroup$ This answer currently reads: the inertial mass is positive because it’s contribution to itself is greater than another contribution. Surely there’s a typo or something missing. $\endgroup$ Feb 9, 2019 at 14:10
$\begingroup$ @ZeroTheHero it is not "its contribution" which is greater, but there is a positive contribution that is greater. But perhaps you are right that it wasn't a good formulation, so I've changed the formulation to make it clearer. $\endgroup$ Feb 9, 2019 at 18:31
$\begingroup$ This is not quite right. Energy is not additive in relativity. The energy in a gravitational field is negative in the Newtonian approximation, but this is not really true in general relativity. For example, the energy of a gravitational wave is positive. $\endgroup$– user4552Feb 9, 2019 at 22:03
$\begingroup$ @BenCrowell that may be for some strong field/heavy bodies where self-energy is substantial or for gravity waves, but I do not think that is the case for ordinary bodies, where gravity energy should be negligible and describable by Newtonian theory. I've edited the answer. Do you think gravitational energy of a 1kg sugar pack is actually positive? $\endgroup$ Feb 9, 2019 at 22:51