Even though my favorite gags on the show involve Dr. Cox, the Janitor, and other characters besides J.D., I think he does hold the show together. He internally narrates almost every episode. The show would be flimsy without that internal narration. I'm sure if Braff left the show, they'd have someone else narrate it.
There have been a few episodes narrated by others. Dr. Cox, Elliot, The Janitor, Carla and I think Turk have all had one episode where they narrate it instead of J.D. If J.D. wasn't on the show anymore, they could just do that. They could alternate narrators.