Things dont fall to the ground because the earth pulls on them. Earth is pulling in the space around it, and those things come with it. See the river model of general relativity for a more thorough explanation.
All of them together are so much stronger it's not even funny. And that's for the "underdense" region that we are in. Not a void, but about half of our galaxy's environment does count as a void.
Gravity ensures that structures at the cluster level and below don't expand as the space they're in expands. The space they're in is expanding just like it is everywhere (assuming a cosmological constant) - gravity just holds them together. Which is what I mean when I say matter isn't pinned to space - it just slides through it.
Gravity is too weak to affect distant objects, so we see the effects of the universe's expansion when we look at them.
i will clarify. in the absence of other forces, matter is indeed pinned to the space it is in.
you talk about gravity as if its something distinct from what is driving the expansion of the universe.
we have that model, its what newton proposed.
Regarding being "pinned," that still fails to account for inertia. The idea that there's a specific piece of space that we're stuck to implies there's a rest state at which there is no motion independent of any observer. We know that's not the case.
My original point was that gravity and the other forces that hold us together are so much stronger than whatever is causing expansion that the expansion of space doesn't affect us at small scales. The space we're occupying is expanding. We're not dragged along with it. The Triangulum galaxy doesn't move away from us because gravity keeps the Local Group together. We do see the expansion of space between us and distant objects, but that's because there's no force strong enough to hold those distant objects to us. That's not because we're "pinned" to our location, but because the space between us is getting larger.
In many interpretations of GR it is believed that mass deforms spacetime, but is also influenced by the spacetime it moves through. In the absence of mass, spacetime expands. Carrying along any mass with it. When mass is present this default expansion is overpowered by its influence which causes a contraction. Hence black holes.
Gravity is still colored by classical definitions so most assume its power is unidirectional. However there is a quantitative factor for expansion in the very same Einstein equation which defines GR.
We need to stop thinking of gravity as a force acting on objects, and rather something that acts on spacetime. Many people advocate this but stop at the rubber sheet analogy, which is completely devoid of the idea that the sheet actually moves as well.
Gravity wells aren't pulling surrounding space toward their centre. They're only pulling other masses that occupy the surrounding space.