No, a complex set of decay processes kills pets; time doesn't do anything. We label things that happen with time to organize causes and effects.

You might as well say that time does everything, but that's mistaking the way you keep track of a thing with the thing itself. Time typed this comment.

The moment you say the cat is "alive" you create time and you also start killing your cat.

In Buddhism there is no birth, no death. Both are illusions.