In classical mechanics, we had Newton's law of gravity $F \propto \frac{Mm}{r^2}$. Because of this, all laws of classical electrostatics applied to classical gravity if we assumed that all charges attracted each other due to Coulomb's law being analogous. We can "tweak" classical electrostatics to fit gravity.
In modern physics, does the reverse work? Can we "tweak" General Relativity to accurately describe electrostatics or even electromagnetism?
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.
First a note of caution: what you are talking about is electro-statics, not dynamics. The reason electrostatics and Newtonian gravitation are similar is that this is really the only possible law that has certain nice properties in 3 dimensions (e.g. Gauss' law). This is of course no definitive proof (the actual proof being based on that these theories are similar approximations of some better theories) but it might give you at least some intuition into the matter.
Now, if you appreciate the point that there might be currents and electromagnetic waves, you must instead consider electrodynamics and then you surely can't "tweak" it to fit Newtonian gravity anymore. Going further, even if you consider gravitational field (as studied in general theory of relativity (GTR)) and electromagnetic field, these concepts are pretty different. So in order to give any answers I have to take your question quite liberally and just tell you what are the similarities between the two field theories and whether they can be modeled together. And for this some answers can indeed be given:
Gravitomagnetism is a linear approximation of the GTR that is equivalent to electrodynamics. This is probably the closest answer to your question there is because in your case you approximate electrodynamics with electrostatics and this is then equivalent with Newtonian gravitation. So here you go the other way although on a much higher level of field theories.
You can consider electrodynamics on the curved background having an effective model for both of them at once. They are completely compatible and in some regards similar (e.g. by having waves propagating at the speed of light).
There is something called Kaluza-Klein theory where you study 5-dimensional space-time in which one dimension models electrodynamics. And, in fact, this turns out to be equivalent to the second point but with an extra field called radion. This is is a baby version of the modern stuff treated in string theory and similar research areas.
Both theories are an example of a gauge theory although general relativity is quite special in this regard.
