The short version is that it depends entirely on the manner in which the moon disintegrates, or more specifically the energy thus imparted on the resultant fragments.
I'm assuming here that the mass remains roughly the same, so nothing like matter-antimatter annihilation. I'm also assuming that the moon is large enough to begin with to become roughly spherical under its own gravitational attraction.
If the moon (or any other celestial body) disintegrates in such a way that the energy imparted on the fragments exceeds the energy corresponding to the total mass' escape velocity, then they will simply spread out. Each fragment will enter an orbit of its own around whatever nearby mass is large enough to allow each fragment to orbit the larger mass at its new velocity. For situations where the difference in mass is large, such as a fragment of what used to be a moon now orbiting a star, this velocity is determined solely by the mass of the larger body. The energy required for this is huge; even a head-on collision with another similarly-massed object moving through the solar system from outside the system might not do it. We have a number of questions dealing with the amount of energy actually required for this; a good place to start might be The opposite to Worldbuilding: World Destruction and the questions linked to and from that one.
If a piece happens to enter an orbit that intersects the planet the moon was orbiting, that'll be a bad day for anyone in its path.
On the other hand, if the disintegration happens with less energy than that required to reach escape velocity from the original mass, then the pieces will remain within their combined gravitational field. Over astronomical timeframes, then, the pieces will re-coalesce into a spherical body, much like how planets form from a protoplanetary disk.