Good question. Keplerian orbits (like the earth around the sun) are stable, according to Newtonian mechanics, gravity will not cause the earth to spiral into the sun. In general relativity an orbiting pair radiate gravity waves, which remove orbital energy and cause them to spiral inward, eventually. The fate of universe appears to be a long slow thermodynamic cool down, with every particle approaching absolute zero, gathered up in gravitationally-bound galaxy-cluster-sized lumps. I’ll miss the old universe.
But there’s a lot we don’t know, so no one really knows for certain.
Actually, for the immediate term (probably billions of years at least) there is no reason that the stars in a galaxy cannot orbit a central black hole. Gravitational radiation would be minimal because of weakness of the gravitational field (the stars are orbiting at large distance). Gravitational radiation would only really be significant for systems with high density objects orbiting each other a close distance (such as two neutron stars orbiting each other, for instance).
Contrary to popular perception, a black hole does not inexorably suck in everything in its vicinity. In fact, for most practical purposes a black hole behaves the same way as any ordinary massive object having the same mass as the black hole. For instance, if the sun were to suddenly turn into a black hole, we of course would die, but not because of being sucked in to the new black hole. We’d die because of the massive ice age caused by the lack of solar radiation, but the earth would continue to orbit the new black hole in exactly the same way as it currently orbits the sun.
The real difference between a black hole and an ordinary object is what happens when you get close to it. The gravitational field of the black hole warps spacetime to such a degree that really wierd-looking things happen to anything that starts to venture too close. For a certain distance, called the event horizon, the spacetime cooridinates change their character, ie time and space interchange, meaning that it is not possible for an object to move away from the black hole’s center. Practical upshot is that an object (or even light) cannot escape the black hole once it gets to the event horizon. Interestingly, an outside observer can never see this occurring. As I stated above, weird things happen near a black hole. To an outside observer, it would take an infinite amount of time for an object to reach the event horizon. (The outside observer would also see the object “smeared out” over the entire surface of the event horizon — told you things are weird near the black hole!)
I despise the notion of an open universe. The very thought of the slow soul-sucking heat death of everything is just absolutely esthetically wrong. I may well, be wrong, but I just absolutely refuse to accept the notion of an open universe. Of course, I doubt I'll get to hang around to see it, but the eventual collapse into a new black hole, that will rebound in a new big bang is a more satisfying prospect IMO.