"The greater the compression, the greater the density" is essentially tautological. Of course packing the same amount of stuff in a smaller space makes it more dense. It's true in the same way the statement "Enlarging things makes them bigger" is true.

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.

Thanks for that reply!

The general theory of relativity does a really, really good job of describing gravity as a result of mass changing the shape of spacetime, and predicting its effects as a result of that. The strength of gravity only depends on the total mass, not its density, so I'd say that gravity is not an emergent phenomenon from the density of matter and energy.

(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.