Yes, on a large scale, the expansion of the universe does lead to an increase in the average distance between galaxies, galaxy clusters, and other cosmic structures. However, at smaller scales, such as within galaxies and within bound systems like atoms, the expansion of the universe does not directly cause objects to move away from each other.
The expansion of the universe is driven by the metric expansion of space, where the space between objects is stretched as the universe expands. This expansion is described by the Hubble's Law, which states that the recessional velocity of distant galaxies is proportional to their distance from us. As a result, galaxies that are farther away from us are receding at faster speeds.
At the scale of atoms and within bound systems, such as molecules or solid objects, the forces holding them together are much stronger than the cosmological expansion. These forces, such as electromagnetic forces, gravitational forces, and nuclear forces, are typically much stronger than the expansion of space. Consequently, the interatomic and intermolecular distances within these systems are determined primarily by these internal forces and not by the expansion of the universe.
In summary, while the universe's expansion causes an increase in the average distances between cosmic structures on a large scale, the internal forces within bound systems like atoms and molecules dominate at smaller scales, keeping the atoms relatively close together.