From looking it up, @Anonymous123 seems to be correct. Some people do seem to lean more towards 歯医者 meaning the clinic itself (in casual contexts). I guess that's more or less the same as when you say "Go to the dentist" in English.
歯科医 = formal, profession-oriented ("a dental doctor") 歯医者 = casual, everyday term (can mean the dentist or their clinic)
I would not say 歯医者 is casual - I'd say it's base level politeness, and the version you're going to hear 99% of the time. I'm actually going to the 歯医者 today, and this is the word I'd use to describe the visit to any person I know.