I don't think so - very few companions die... Although one of them attempted to kill him at one point - in "The Wedding of River Song" River shoots at who she thinks is the Doctor, but is in fact the Tesalecta.
Yes, the Fifth Doctor shot and destroyed the robot, Kamelion in "Planet of Fire" (at Kamelion's own request, to end what Kamelion considered unbearable suffering). He has also had several other companions die during travels if that counts.
- the Doctor also technically killed Sara Kingdom (whose status as a companion really varies depending on who you're talking to), as she was caught in the time destructor after the Doctor activated it. If you really want to be cynical, the Doctor could also be blamed for killing any companion who dies under his watch if he invited them to travel with him.
- The question asks about any he killed, not about any for whose deaths he could by some stretch of the imagination be thought responsible. It's arguable that he's not killed any. Kamelion was a robot. However, the Doctor certainly did destroy him but (as indicated above) that was as a requested act of mercy.