If we were being poetic, here, then I'd say that Man's greatest enemy is Man themselves, but this is HN, so I won't. For a runner up, I would at least choose an animal that can actually kill a healthy human, rather than one that and annoy a healthy human and kill a subset with allergic reactions of venom.
How about one that kills one to three million humans, mostly very young children, every year? The mosquito is man's greatest animal enemy. Actually, I think it's safe to say that mosquito is a worse enemy to man than man himself, since mosquito-borne malaria has been responsible for many more deaths than all the wars in history.
Big predators, on the other hand, like bears or lions, which are physically capable of killing a human single-handedly, are responsible for so few human deaths as to be lost in statistical noise.