Harbour island has clinics but no hospitals. The closest hospital is in Nassau which is a 30 minute flight from the Airport of Eleuthera. There are way more flights from North Eleuthera to South Florida than to Nassau. We therefore suggest that you fly back to the US to get to a hospital, you probably have medical insurance and will be closer to home.
If it is a real emergency you should be repatriated by a flying ambulance hoping that you it is included in your insurance package.