Could the gravitational force be what holds the charge of the electron together? It seems to be the only obvious possibility; what other ideas have been proposed besides side-stepping the issue and assuming a "point charge"? How would this affect the electron "self-energy" problem? The question is related to the idea of geons.
This was an idea Einstein had soon after developing GR, and it is developed in the classical unified field literature, with the starting point being Einstein's "Do Gravitational Fields Play a Role in the Constitution of the Elementary Particles?" This is one of the defining program papers of the unified field framework.
This idea is also discussed off and on within heuristic models of charges throughout the 1950s-80s. The essential points are that as you make the electron smaller, at some point, gravity will become dominant. The problem with such ideas is that they generally do not have a good idea of quantum gravity to make the microscopic model precise.
All these classically inspired ideas are obsoleted by string theory and subsumed into it. Within string theory, the fundamental objects are dual to black holes, so that their classical limit is identifiable as a recognizable extremally charged black hole of the classical limit supergravity theory. Aside from identifiable black holes, there is no other matter (arguably--- there is the question of whether orbifolds count as "matter"). So for example, for the M-theory, the objects are the extremally charged M2-branes which are the extremally charged black holes you can make using the 3 form gauge field, and their 5-brane magnetic duals (the magnetically extremally charged black holes). That's it for M theory. The brane-spectrum of a theory is the answer to classical question "what extremal black hole can I form?"
The identification of black holes with matter is important, because the internal construction of strings is fully specified by the theory. So that the electron, if it is a string theory excitation, is an object whose internal structure is completely known, because you know the scattering off the electron at arbitrarily high energies. Further, in the strong scattering case, we can continuously link the electron to both netural and extremally charged black holes, so that the theory is a full realization of Einstein's program.
Since I believe string theory is the correct theory of everything, I don't think there is much point in investigating these types of ideas in a different direction. But some people who like loops disagree.
Yes, it can. Here is a toy model using Newtonian gravity.
V/c^2 = e^2/mc^2r - /\r^2
e^2/mc^2 = rc (classical electron radius)
with SSS metric
g00 = 1 + 2V/c^2 = - 1/grr
g/c^2 = -dV/dr = + rs/r + 2/\r
We can get g = 0 with /\ < 0 i.e. AdS metric
In a vacuum where the w = -1 virtual electron positron pairs surrounding the bare charge have higher density than the w = -1 virtual photons, we can have /\ < 0.
The equilibrium will also be stable looking at d^2V/dr^2.
This neglects spin, but we can model that with the centrifugal potential
The short answer... we do not know. (Were 'we' is humanity or physicists - take your pick.)
A more interesting answer is... The electron size is known to be 10^-18 meters or smaller. If gravity was holding it together then it might be at the Schwarzschild radius.
Rs = 2GM/c^2
so with values substituted it would be
2 x 6.67300x10^-11 x 9.10938291x10^−31 / (3x10^8)^2 = 1.35x10^-57 meters
However, this is less than the Plank length (10^-33 m). Therefore, if it is held by gravity then it would likely have a radius near the plank length. Supersymmetry (SUSY), for example, has gravity increasing to have all forces equal at the plank length.
If you check out Lenard Susskind's lectures on ER=EPR and GR you will find that he thinks that physics is leading us towards the idea that elementary particles and black holes might be related. Black holes have only three properties angular momentum (spin) mass, and charge. Sounds like an elementary particle.
This is a hint not a theory. It is very early to say.
EDIT: It would have been nice to know why this was downgraded with the addition of a comment. There is nothing wrong with the physics. If it was downgraded because it does not answer the question then that does not make sense because there is no known answer to the question.
Black holes look like macroscopic elementary particles but elementary particles do not look like black holes.