Once you have pushed all galaxies in the Universe so they move away from each other, you do not need any special forces to keep their motion away from each other going (following Newton's first law), given that space is nearly empty and doesn't yield much friction.
One answer I've read to the question "why does the Universe keep expanding?" is "because it did so in the past". No currently active special force to stretch everything, including galaxies, is needed. Of course, this does not explain the process that started the expansion in the first place, but does explain how it can keep going.
If a special stretching force does exist (and it may, given that the expansion of the Universe appears to be accelerating), then it could only cause a galaxy to expand if that repulsive force were strengthening all the time (over the same distance), compared to the forces that keep the galaxy together (chiefly gravity).
Otherwise the forces within the galaxy (including the repulsive force) would on average keep the same balance.
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