No you dont have to have a pilot certificate to be an air traffic controller. I am currently at UND and every ATC person has to get some flight experience. You can either take a survey of flight class or get your PPL. I got mine becuase I thought it would be cool. And yes you do have to get a medical to be a controller and from what I hear it is like a 2nd class medical.
Nightmare 68, Fargo Tower, Runway 36, Fly Runway Heading, Mantain 10,000, Cleared For Takeoff, Change To Departure