will make the acceleration asymptote to zero? Also, gravity is a very weak force that drops off quickly. At a finite escape velocity space probes like Voyager 1/2 (ignoring outside factors), will never reach equilibrium distance with the sun, they will keep moving away indefinitely. Likewise I don't think what you suggest would cause the expansion to stop, though acceleration of that expansion might. Remember that no net force means no change in momentum, not that things are static.

This is all pretty hand wavy, a proper answer would require calculations from general relativity.