The answer to this question is "not really."
The gravity around a black hole remains normal unless you get extremely close. If the Sun suddenly became a black hole (which isn't actually possible), the Earth and all the other planets would continue to orbit it just as though nothing had changed.
The behavior of gravity doesn't change until an object approaches to the point where it's within a few times the radius of the event horizon, the boundary marking the region around a black hole from which not even light can escape. At that point, objects begin to lose the ability to maintain stable orbits, and inevitably spiral into the black hole.
So to return to our theoretical example, if the Sun became a black hole, objects would have to be as close as about 6.2 miles (10 km) to the black hole's center before they began spiraling in.
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