I think you mixed up galaxies moving away from each other, and a galaxy’s stars etc. moving in space.

As per Einstein’s relativity theory, gravitational force has infinite range. So there will always be some pulling force between galaxies, which means they would eventually slow down and and eventually start moving towards each other. But our observations suggest that they are moving even faster day by day. So there must be some force that is stronger than gravity and it must be somehow pushing objects.

So gravity by itself doesn’t explain the speed increase of universe’s expantion.