Yes, any impact big enough to destroy one of a pair of twin planets would render the other uninhabitable.
The way the question is asked, it appears that the destroyed plant is not merely rendered uninhabitable, but is actually completely disassembled. (You talk of planet debris possibly forming a new moon.) Additionally, the planets would appear to be roughly the same mass and big enough to hold atmospheres.
The energy required to do disrupt a planet would not just pulverize it, but would sent bits of all sizes flying in all directions. The undisrupted twin would get a large amount of matter hitting at just under escape velocity. The upshot would be conditions that would make the K-T impact look like a picnic. It is very unlikely that any life would survive on the surface.
Without detailed simulations, and without knowing the direction, speed and mass of the impact I can't estimate the amount of material impacting the twin, but it would probably be large. (It would be huge if the impact was "head-on" and the twin lost a lot of its orbital angular momentum.)
Having 1% of the mass of an earthlike planet rain down on Earth at 80% of escape velocity would probably melt most of Earth's surface, though it would not disrupt Earth. (You can get this estimate by remembering that 5 miles/second turned into heat is enough to melt the impacting body. Adding 1% of the Earth's mass in lava to the Earth's surface all at once makes a pretty thorough mess.)
In most scenarios, a substantial amount of material would remain in orbit forming a disk which would in turn form one or more new satellites. (You can actually get a very good idea of what happens by looking at the scientific literature about the formation of the Moon due to a giant impact on Earth.)
Bottom line: If you live in a twin planet system, do not use a giant impactor to disrupt your twin planet, no matter how much they annoy you.