The expansion of the universe is currently accelerating faster than gravity can slow it down, so at very large scales, galaxies are being pushed apart, not pulled together. In that sense the universe is going to continue to expand indefinitely, not get eaten.
However, in the very distant future, there is likely to come an era when black holes dominate the universe because, as you’ve guessed here, they’ve pulled in all the available matter around themselves. The last generations of stars will be dying out, there will be fewer and fewer gas clouds left to collapse into new stars, and so the debris of dead stars — like black holes — will dominate the universe.
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