My cat caught a wild bird but I saved it (on accident) and it doesn't appear to have any damage but can't fly.?

You most likely have a fledgling. Fledgelings are baby birds who are learning to fly. Characteristically they are identified by their round bodies and soft feathery features. If you let that bird out into the wild, it will die. When fledgelings fly out of their nests and fall onto the ground, they may seem to be in danger but their mothers are ALWAYS close by to feed it and make sure it's not in peril. The moment you touched that bird, you put it in danger because the mother will no longer look after it after a human has come into contact with it. I would keep the bird until it can fly, but be careful with your cat when it does. Let it free once it has learned to fly.