No, they won’t. Two of Jupiter’s prograde moons will:
http://www.seasky.org/solar-system/jupiter-moons.html
The retrograde moons and numerous retrograde moonlets may eventually crash into Jupiter, or they may be ejected from the system.
Phobos will eventually crash into Mars, Deimos probably won’t. They must have been captured by Mars, but no one has come up with the math to satisfy everyone else working on the problem.
http://nineplanets.org/phobos.html
http://nineplanets.org/deimos.html