The parallel between Gravity and E&M is that both forces are mediated by massless particles, the graviton and the photon, respectively. This, in the end of the day, is the reason why both classical theories look similar.
But, when you really study what's going on behind the scene, you learn that Gravity is more appropriately described by General Relativity (GR) and ElectroMagnetism is better described by Quantum ElectroDynamics (QED).
The resemblance of these two theories, one may say, rests in the fact that both are described by the same mathematical framework: a principle bundle. In GR's (ie, gravity) case this bundle is a Tangent Bundle (or an $SO(3,1)$-bundle) and in QED's (ie, E&M) case it's a $U(1)$-bundle. The geometric structure is the same, what changes is the "gauge group", the object that describes the symmetries of each theory.
Under this new sense, then, your question could be posed this way: "Is there a way to modify geometry in order to incorporate both of the symmetries of these two theories?"
Now, this question was attacked by Hermann Weyl in his book Space, Time and Matter, giving birth to what we now call Gauge Theory.
As it turns out, Weyl's observations amounts to a slight change on what symmetries we use to describe Gravity: rather than only using $SO(3,1)$, Weyl used a different group of symmetries, called Conformal.
As Einstein later showed, it turns out that if you try and describe Gravity and E&M using this generalized group of symmetries (under this new geometrical framework of principle bundles) you do not get the appropriate radiation rates for atoms, ie, atoms which we know to be stable (they don't spontaneously decay radioactively) would not be so under Weyl's proposal.
After this blow, this notion of unifying Gravity and E&M via a generalization of the geometry (principal bundles) that describes both of them, was put aside: it's virtually impossible to get stable atoms (stability of matter) this way.
But, people tried a slightly different construction: they posited that spacetime was 5-dimensional (rather than 4-dimensional, as we see everyday) and constructed something called
a Kaluza-Klein theory.
So, rather than encode the E&M symmetries by changing the geometry via the use of the Conformal Group, they changed it by increasing its dimension.
Now, this proposal has its own drawbacks, for instance, the sore thumb that is supradimensionality, ie, the fact that spacetime is assumed to be 5-dimensional (rather than 4-dim) — there are other technicalities, but let's leave those for later.
The bottom-line is that it's proven very hard to describe gravity together with the other forces of Nature. In fact, we can describe the Strong Force, the Wear Force and ElectroMagnetism all together: this is called the "Standard Model of Particle Physics". But we cannot incorporate gravity in this description, despite decades of trying.
This post imported from StackExchange Physics at 2014-03-24 03:26 (UCT), posted by SE-user Daniel