[riperoni]

Well, not really.

The expansion of the universe is not responsible for explaining gravity, it was observed and aided in determining models and constants for the theory.

Start by looking at a particles property called mass. Mass interacts with the higgs field by exchanging a particle of information called the higgs boson. The interaction strength, corresponding to mass, determines how much the space time around our particle curves. Space time curvature dictates a particles movement. Now throw another particle with mass next to the initial one, also interacting with the higgs field. This is when you get into a huge loop of interacting movements and fields.

Locally and classically we approximate it by the classical law of gravity which loosely states that two masses attract each other, depending on how close they are, i.e. GMm/r^2

The universe expanding or contracting is a consequence of mass and gravotational fields, not the other way around. And calculating the expansion and thus curvature of the universe gives you a picture of the average mass distribution.