Depends on the distance… which means, probably not.
But keep in mind that "blowing up" a planet doesn't remove the mass. It just gives that mass a briefly energetic acceleration away from its own center of mass (that would be quite the explosion!)
But notice that this would be instantaneous acceleration, not sustained. The mass itself is still the same and still has the same attraction to its constituent bits. As a result, they would all begin to decelerate towards each other from the instant of the explosion. So then it becomes an issue of how much kinetic energy was given to the various parts by the explosion vs. how much (at what rate) the gravitational attraction will overcome that brief application of energy.
It might entirely be possible that the acceleration is minimized— only what was needed to universally "crack the crust," which would be quite sufficient to kill every living thing on the planet, after which it all "falls" back together again.
But as can be seen on film, the Empire is not a big believer in conservation of energy, or efficiency in destruction (minimum effort for maximum casualties). It would appear that they went for full disruptive separation of the agglomerated planetary matter, applying enough energy to fracture the planetary core into relative small particles— small enough that even the Millennium Falcon's deflector screens and hull can essentially handle the various bits. That's some serious overkill.
On the other hand, at the time of the Falcon's arrival from hyperspace, the particulate elements of Alderaan are not moving at any great speed, if the screens and hull plates can treat the impacts so casually. The assumption here must be that the mass is already falling back in on itself, and that it is the Falcon's relative speed which is causing the impacts.
Also, the fact that it's stated that "it's not there," means that the hyperspace window for arrival is relatively close to the planet's actual location, and that the "asteroid belt" is thus the planet's own matter, which is already slowing, or in a moment of relatively little momentum, or possibly even already reversing its acceleration back towards all the mass's combine center of gravity.
If that is the case, then any disruption to the orbit of other planets would be minimal, if even registering. As far as these distant bodies are concerned, the entire gravitational effect of Alderaan is still fully present in essentially the same place and orbit it always was.
So give the Empire credit for carefully planned, selective destruction with no environmental impact on distant planets or their system.