I assume you know that medical care on island is not going to be up to US standards. Hopefully you purchased some trip insurance which covers medical issues.

cruzer answered your question regarding antibiotics. I think that's the same way, both sides, a pharmacist can fille some things without a scrip, that would require one in the US.

The French side hospital is Louis Constant Fleming Hospital, and it is in Marigot, although I have no idea where.
The Dutch side hospital is close to Belair and P'burg.

I only have had experience with the Dutch side hospital and wasn't impressed with their ER department, although it was several years ago.

I have heard from some locals that some folks on the Dutch side use the French side hospital, as they consider the French side hospital better, but I personally wouldn't know.

Hopefully you won't have any need for such things on island!