Is it possible that the force due to gravity varies based on density? It took a while to find relativity and the various subtle (or unmeasurable) effects it has within our universe.  Is it possible that, although Newton's gravitational force equation leaves out any notion of density or size, this too might impact the forces acting around us?
I noticed this question already, but I am not satisfied with its answer.  I wonder if two objects, A and B, having the same mass but (for lack of a better term) relativistically different densities, would exert slightly different gravitational forces on a third object C which is far enough from A and B (and equally distant from both) to negate any "diffusion" effects on the gravitational fields of A and B?
In other words, is it possible that the equations we have are "good enough", and we just don't have examples (yet) that introduce the same kind of errors as those that confirmed the theory of relativity?
If not, why not?  Is there any particular reason aside from "these are the equations we have"?
I realize that finding the mass of something like a black hole would be at best impractical, if not entirely impossible, without assuming that its mass directly determines its gravitational field strength.
 A: The answer is No and the reason is the equivalence principle which says that there exist natural units in which the gravitational mass (the mass $m$ in $F=GMm/r^2$) is equal to the inertial mass (the mass $m$ in $F=ma$) for all objects in the Universe. This is equivalent to the statement that all objects, regardless of their composition, density, and other properties, accelerate by the same acceleration in a given gravitational field (any gravitational field).
This equivalence principle is the starting point behind Einstein's general theory of relativity which describes gravity as the curvature of spacetime (and which is fully respected by more modern theories of gravity, especially string theory). This principle is also experimentally verified at the relative accuracy level of order $10^{-17}$ which is incredibly accurate (one may compare two different materials which have noticeably different densities etc. and the acceleration is still perfectly the same). In principle, it's always possible that some experimental deviations will be found by finer experiments in the future (but that's true about any insight in physics). However, it's not just the absence of any experimental hints that undermines any idea that the gravitational force should depend on anything else aside from mass; it is also the complete absence of candidate theories that would be compatible with the basic observations and where the deviation would be implanted as anything more than just an ugly, partially inconsistent, unjustified, and numerically small deformation of a beautiful, consistent, robust, justified theory.
A: It depends on the shape of the objects A and B and the distance from them. The Jet Propulsion Laboratory released a free iPhone and Android app called "Earth Now". If you install it and run it there is a feature called Gravity. It depicts the different gravity strength in different regions on the Earth, and those are slowly, but constantly changing even though overall they remain pretty much the same over the year. If you observe, you will see slight differences from time to time. 
   However, this doesn't mean that gravity depends on density, but solely on the mass. As we all know inside of the Earth there's a flow of hot matter, and it has of course different density and different mass in different regions inside and they all are moving around, even though the movement is slow. I'm sure you heard about movement of continents, and you can understand thus that gravity depends on mass, even though it looks like it depends on density. More dense objects have a bigger mass, it all depends on the material. So, if A and B were perfectly spherical objects and if the distance from them is very big, then there's no way that density which is same throughout the entire object can influence gravity. However, if the bodies are irregularly shaped and the distance isn't so great, then there is a possibility that the sum of regional masses on A is different than the sum of regional masses on B and they will exert slightly different gravity to the point C which is not that far from A or B. Draw two circles not far from each other and place the point C between them and draw lines between point C and different regions with same density, and make an assumption based on your findings.
A: This is a real question. I don't know the answer to it. Some people seem to have the strong intuition that the universe actually would follow all the laws of general relativity including the law that the gravitational field of any two pieces of nonrotating stationary matter of a given mass will always asymptotically approach each other as the distance approaches infinity if the cosmological constant were zero. I don't see why that must be the case. In the past, some people were very stubbornly insisting on the truth of an assumption that turned out to be wrong. Just because we have not yet observed an object have a gravitational field strength different than its mass times the accepted gravitational constant doesn't mean it can't ever happen. Maybe a neutron star is made of a different substance so it actually has a gravitational field strength different from its mass times the accepted gravitational constant. If we can't measure its mass in another way than its gravitational field strength, how do we know that actually is its mass?
