Tukukkk

Consider a simple universe consisting of two very distant galaxies (neglect gravity between them). The relative motion between them is such that, if the universe were not expanding, they would be at rest relative to each other. Due to the expansion of the universe, however, the distance between them keeps increasing, at an accelerated rate, suppose.

My general question is: is this "acceleration" literally a kind of (accelerated) motion?

It surely is similar to literal motion in many respects. It can be measured in the same units as literal acceleration. Light reaching each galaxy from the other is suitably red-shifted... Etc.

Yet, it is not like literal motion in other respects. No energy is needed to accelerate either galaxy (I take it that dark energy is postulated only to overcome gravity, not to "cause" the expansion... right?). Relative velocity can (and actually does in some cases) exceed the speed of light, without contradicting SR. No inertial forces are thought to be involved due to their "acceleration" (or are there?). No repulsive force between the galaxies could possibly "cause" this "motion" (at most, a repulsive force would cause the two galaxies to accelerate, in the literal sense). Etc.

So, is it or is it not literally a kind of motion? For example, were the expansion of the universe suddenly to stop, would the two galaxies continue moving away from each other at a constant speed (as they would if it was literally a kind of motion)?

The universe is expanding overall, and galaxies have relatively small “peculiar velocities” relative to the expanding universe. If you ignore the latter, then if the universe were to stop expanding, the galaxies would stop moving apart.

So the “motion” of galaxies due to the expanding universe is quite different from ordinary “inertial” motion. There are no non-inertial forces arising from the accelerating expansion of the universe.

You really should not think of the galaxies as moving at all. (Again, ignoring their “peculiar motion”.) They are sitting still in an expanding universe, while expanding space causes the distance between them and other galaxies to increase.

Since they are not really moving, this explains why their separation can increase at a rate faster than $c$. There is no speed limit on the expansion of space. There is only a speed limit on how fast objects can move through space.

My general question is: is this "acceleration" literally a kind of
(accelerated) motion?

That depends on your definition of various terms like distance, speed and acceleration. Generally the answer is "yes", this is "real" acceleration. And that's because relativity told us not to confuse measurements with something "outside" the universe, whatever you're using to measure that acceleration is part of the universe too.

But I think you may actually be asking another question, so let me expand a bit...

What I think you're asking is whether the galaxies are accelerating relative to the universe. That is, if the two galaxies are at rest relative to each other now, and they are not in the future, then is that because the universe itself moved, or the galaxies within it?

In that case we need some way to measure that original speed. We have such a way. If you consider the Cosmic Background Radiation to be at rest relative to the universe, then you can measure your velocity relative to it. If you did this before and after the galaxies accelerated away from each other, they would both notice no change - so in that respect they didn't accelerate, and it was the universe itself did.

Whether there's a difference between those two statements really comes down to various definitions. But back to the original question...

If the expansion of the universe were to stop, would galaxies continue
moving away from each other at a constant speed?

Yes, at least in relation to their existing speed relative to the CMB. For the Milky Way that;s, IIRC, about 0.01 c.

Whether you say that the universe is expanding, or that the galaxies are moving away from each other, is just a matter of what reference frame you choose to use. Both descriptions are equally legitimate.

If you imagine a rigid grid with uniform spacing in all spacial directions, then it is natural to say that the galaxies are moving. If you imagine a flexible grid with galaxies at fixed coordinate values, then it would be natural to say that the universe is expanding.

In principle, General Relativity applies to either type of frame. However, it has no clear way of comparing times, speeds, and distances of very distant objects, unless particular reference frames are used. The Friedman equations that describe the expanding universe use a particularly convenient frame with fixed coordinates for the galaxies and a notion of an overall cosmological time, together with a time-varying scale factor for the grid. But that is just a computational convenience, not a conceptual imperative.

So, your question as written doesn't really make sense. Stopping the expansion would mean stopping the motion. And the question of whether the expansion is accelerating or not makes no difference in this regard.