What if gravity is an emergent result from the compression of matter? The greater the compression, the greater the density and gravity?
But as for gravity, no. Gravity is a function of mass. Compressing a body into a smaller space will not increase the gravity you experience at a given distance from the mass's center of gravity (though it will allow you to get closer to the center of gravity). If you magically replaced the Sun with a black hole of equal mass, the planets would continue their orbits undisturbed — even though a black hole's singularity is infinitely dense and the Sun is less dense than the Earth on average.
(Mass is almost certainly caused by the interaction between some quantum particles and the Higgs field. I'm not clear on how that deforms spacetime, though.)
Edit: I should refresh the page before posting.