Theoretically could the expansion of the universe affect acceleration due to gravity? I was wondering if theoretically the universe expanding affects speed or acceleration due to gravity, compared to if space weren't accelerating. 

The way I have it in my head, something is falling towards a planet with gravity G acting upon it. It's been falling for T amount of time and has gone X distance with Y left to go. Currently the universe is expanding, so the distance between its starting and ending point is a very tiny bit longer when it arrives than when it started, either affecting its distance traveled at time T, or its speed at time T.  Compare this to a non-expanding universe where the distance between start and end remains the same, no matter what point in time you choose. 

I know that local gravity forces and atomic forces greatly overpower expansion forces so galaxies, solar systems, planets, and watermelons don't just fly apart. I'm just wondering if the value of either gravitational acceleration, or an object's speed could theoretically be changed slightly due to cosmic expansion.
Right now the expansion of space is only observed on scales like that of galaxies and the entire universe. We say that with the fancy term globally. Expansion of space does not take place within galaxies at least for now.
As pela has rightfully noted in the comments, for now the gravitation of galaxies, stars etc. is strong enough to overcome the expansion of the universe. But as General Relativity tells us gravity is local, that is, it only affects objects that are considerably close. So, on large scales expansion continues. (I am no expert, but I think that the reason for expansion not happening within galaxies out smaller systems is that the matter and energy is more dense, resolution by in stronger gravitational fields).
It is hypothesized that the universe may end with a big rip: this means that the expansion eventually would start to affect galaxies. Galaxies, star systems, planets and eventually atoms will be ripped apart due to the expansion of space. When and if that happens, not only gravity but not even electromagnetism or the strong force would work to overcome the mighty expansion.
But this situation is highly hypothetical and maybe it won't even happen. So for now, the answer is no: the expansion of the universe could not affect gravitational acceleration.