This is not settled physics by any account. However, according to the Scionics Institute, something sort of like what you described is actually what does happen, but without the formation of black holes. This can be explained using a layman's description of virtual particles (which should be sufficient to get the point across here here):
Virtual particles which emerge at the edge of an event horizon can become permanently separated due to one falling into the black hole and one escaping out into space, thereby becoming a "real" particle. Similarly, as the acceleration of the universe continues to increase, it also becomes increasingly difficult for pairs of virtual particles to mutually annihilate, as they get pulled away from each other due to the super-fast (and ever accelerating) expansion of space itself. (This is pretty much what you already said, I'm just repeating for clarity.)
Eventually, as enough virtual particles are converted into real particles (or real energy, as the two are equivalent) their mutual gravitational attraction halts the accelerating expansion. At this point the universe is in a state exactly as it was at some point after what we commonly call the big bang: An extremely dense universe, filled with energy (or mass, as again, these are equivalent) which had just undergone an exponential expansion, but is now expanding linearly, but still VERY fast, fast enough for virtual particles to keep coming into existence and becoming "real." This continues until the accumulation of real particles slows the expansion to the point where virtual particles can no longer be pulled apart and become real. Still super-fast, but just not fast enough for virtual particles to become real.
So, the "end" of our observable universe (not the whole universe itself, just the part we can observe) looks pretty much exactly like the "beginning" of our observable universe. In this model, then, the entire universe (not just the part we can observe) is actually infinite, eternal, and eternally expanding, albeit at different rates at different times and places.
There is no actual beginning or end to the universe. 781
It is eternal. There are, however, different "epochs" marked by what we think of as "big bangs" and "big rips" - which are actually the same thing! (And really, the "big rips" in this model never actually get to the point of eternally "ripping" everything apart, because it is stopped by the appearance of real particles everywhere.)
Our observable part of the universe was once smaller than an atom, but has now expanded to the size it currently is. (This expansion was super-fast during the big rip/bang phase, but is now much slower, although slightly accelerating at this point.) At the end of this epoch, every part of the observable universe which is the size of an atom will itself expand to the size of our observable universe, and bigger. (Actually, ultimately infinitely.)
This is definitely a ground-breaking view which is new enough that it has not been widely reviewed or accepted as yet. But I think it will be. It avoids the postulated big-bang singularity, and the logical inconsistency (i.e., physical impossibility) of having a singularity containing the mass-energy of the universe explode. It also avoids having a universe (or existence itself) spring forth "from nothing." The universe always was and always will be, and always was and will be expanding (again, at different rates at different times and places). And while it might be hard to imagine the expansion of something infinite, it makes perfect mathematical sense when expansion is viewed as the the creation of new points (or Planck units, in the case of the physical universe) coming into existence between preexisting ones.
This is laid out in an article or book called "Matheism and Psychonics" but the specific part about cosmology and the big bang/rip is here. 781