If an object is in free fall its trajectory is a 'geodesic'. In this case the object doesn't feel a force acting on it. This is true for Galaxies, planets orbiting around a star and apples falling down to Earth.

As @timdeeg has pointed out, this does not fit with observation because your model predicts that galaxies should feel a force, and they don't.

The reason the concept of "space expanding" doesn't make sense is that "space" is frame-dependent, and the laws of physics are frame-independent. So whatever is going on in the universe, "space expanding" can't be a good description of it.

I was using ‘force’ in very loose terms just to indicate there was something going on to cause objects to move apart. I wasn’t implying they felt any force from acceleration. The way I thought about it wouldn’t be any different.

Then you shouldn't use the word "force", because it will only lead to confusion. Not just for others, but for you. It leads you to think that "there was something going on to cause objects to move apart". There isn't. There is just the geometry of spacetime.

To illustrate what I mean, consider tidal gravity: two objects free-falling radially above a gravitating mass (like the Earth), starting from slightly different altitudes. These objects will move apart as they fall. Is there "something going on" that causes this? If so, what is this "something"? If not, how is this case different from the expansion of the universe?

Well, to know what geodesic means is not the clue to have a notion of what expansion means. But you can imagine two neighboring geodesics describing the trajectories of two objects. If the spacetime is curved then their geodesics accelerate relative to each other, in the case of an expanding universe they accelerate away from each other. Whereas if the spacetime is flat their relative acceleration is zero (which doesn't exclude of course that these particles move relative to each other with constant speed).

This is not quite right. In a matter-dominated universe the expansion is decelerating, and the geodesics in question (the worldlines of comoving objects) are converging, not diverging. But the universe is still expanding.

The correct definition of "expanding" for the universe is that the congruence of worldlines of comoving objects has a positive expansion scalar. Unfortunately, that's already getting beyond the "B" level of this thread. But you can find more information here:

The expansion is due to the positive expansion scalar of the congruence of comoving worldlines (see my previous post). But the fact that such a congruence exists and has the properties it has (not just positive expansion, but every comoving observer sees the universe as homogeneous and isotropic) is due to the particular geometry of the spacetime in question (FRW spacetime).

Ah yes, it follows from the second Friedmann equation that the second derivative of the scale factor is negative in this case (what I didn’t take into account). Thanks for correcting and for the link.