Black holes aren’t special in terms of how their gravity pulls on things, they’re just special because they’re very dense so the force of gravity on their “surface” is extremely high.
The Earth could be a black hole if it was all compressed down to a little smaller than a centimeter across. If that happened the moon and all the satellites orbiting the Earth wouldn’t even really notice – from their orbit the gravitational pull of the Earth is the same, the only difference would be that light can’t escape from the surface of the Earth anymore.
So really the reason why black holes don’t destroy the universe is the exact same as why the Earth doesn’t destroy the universe, or the sun, or any object in space. Everything is moving around really fast, and even though they’re pulling on each other through gravity the force they’re pulling with usually just isn’t enough to really affect things that don’t happen to accidentally pass really close on their own.
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