If gravity is this "unexplainable force" that pulls everything to the center of a planet or stellar remnant you stand upon, why doesn't gravity pull itself?
If gravity affects anything with energy, why doesn't gravity affect itself?
Gravity is energy, right?
Gravity does affect itself. What you've just given is basically a correct argument for the fact that general relativity must be a nonlinear theory.
Gravity is energy, right?
Gravity has energy. For example, the mass of the earth contains a negative contribution from its negative gravitational binding energy.
In more detail, here is an argument to the effect that general relativity must be nonlinear.
As a concrete example, when the earth condensed out of the primordial solar nebula, large amounts of heat were produced, and this energy was then gradually radiated into outer space, decreasing the total mass of the earth. If we pretend, as in the figure, that this process involved the merging of only two bodies, each with mass $m$, then the net result was essentially to take separated masses $m$ and $m$ at rest, and bring them close together to form close-neighbor masses $m$ and $m$, again at rest.
The amount of energy radiated away was proportional to $m^2$, so the inertial mass of the combined system has been reduced from $2m$ to $2m+\delta$, where $\delta\sim -G/c^2r$. The reduction in inertial mass due to radiation in this scenario is in fact almost exactly identical to the result of the thought experiment used by Einstein in his original paper on $E=mc^2$. Based on the equivalence principle, we expect that this reduction in inertial mass must be accompanied by an equal reduction in the gravitational mass. We therefore find that there is a nonlinear dependence of the gravitational field on the masses.
This nonlinearity is incorporated into general relativity in the Einstein field equation, which is a nonlinear differential equation.
Gravity, the physical phenomena, does affect itself. On the other hand the physical theories expected to explain gravity may of may not admit this self interaction. The General Theory of Relativity does admit while Newtonian Gravitation does not.
Newtonian Gravitation satisfies the Superposition Principle: If you have two masses $M_1$ and $M_2$ attracting a test particle $m$ the resultant force on the probe is just the vector sum of the forces each of the masses $M_1$ and $M_2$ generates separately. The same does not happen in General Relativity. In Einstein's gravitation, energy also generates gravity. The two masses $M_1$ and $M_2$ interact gravitationally with each other which means this system have energy. This energy generates an additional gravity on $m_1$. The spacetime deformations generated by each mass isolated do not simply add when we put them together. In other words the resultant force on $m_1$ is different from the sum of the forces generated by $M_1$ and $M_2$ separately.
There are alternative ways to understand this "lack of superposition" presented by gravity and admitted by General Relativity. Microscopically we can think of gravitons which self interact. This interaction leads to energy which leads to more gravitons. You get the idea. Notice this effect does not happens with electromagnetism. The photons do not interact with themselves (at least at low energies) and electromagnetism satisfies the Superposition Principle.
Other people prefer to think about fields and their symmetries. In this case one has theories which are locally symmetric under certain groups of symmetry. For General Relativity this group is non-Abelian (two consecutive coordinate transformations depend on the order). The same happens for QCD. This non-Abelian property implies in non-linear equations which (from the mathematical point of view) can be seen as the reason why the Superposition Principle fails.
Within the General Relativity formulation of the gravitational field , yes the field does interact with itself. Which makes the model of it highly non-linear. One of the reasons there is no viable quantum theory of gravity.
Note - gravity is not a pull in GR.
Anyway, I think the answer is no. If that was true, wouldn't all masses turn into black holes.
If gravity pulled itself (because it is energy), then it would create more energy by pulling on itself, thus more gravity and so on.. eventually turning into a black hole.
Gravity pulls on itself only in the sense that the pull extends to large distances. Just like a large mass less string was used to pull.
