My cat killed a bird - do I need to take him to the vet to make sure he didn't get sick from it?

No - I'm sure its not the first bird he's killed and probably has eaten others. If he's up to date on shots, he's fine. But you may want to rethink letting him out unsupervised all the time. Consider training to a harness/leash and go out with him or build/buy a cat enclosure for him to sit in and be safe (and not hunt birds, etc).