Superman killed both Zod and Doomsday in the comics because he had no other choice and those actions haunted him forever. In the movie is also obvious he had no other choice and he was incredible affected by what he had to do.

I have to rewatch Superman II again because I don't remember exactly how Zod died but If I remember correctly he fell somewhere so people doesn't seem to mind Superman pushing someone to the abyss because he doesn't really get his hands "dirty".