Incomplete Sentence- Notre Dame?

I would be tempted to re-write by saying: The answer is "a Catholic university with an excellent academic reputation." You are correct that the original is not a complete sentence because it has no verb. But you also have the right idea in using your own "voice." Your original sounds conversational and natural, which is a quality you want to keep. My re-write keeps your "voice" but adds the verb "is" to create a complete sentence.