I dunno... it'd be a close match, if that cat has claws and any hunting or fighting experience. If he damages the eagle's wings, it's only a matter of time before the cat kills him. If the eagle took off and tried to grab the cat, he'd soon have to drop it, but i guess the cat would lose by default in that case, depending on the altitude...

Our family cat, when I was a kid, once nearly killed a seagull... their claws aren't as bad as an eagle's, but they can do some damage with that beak... the gull barely escaped alive, and the cat didn't have a mark on him.