They can prescribe a drug which is not FDA approved, can they?
You also have the question of which drugs need a prescription or are considered a controlled substance, which is certainly government regulation regardless of whether or not the regulatory agency is the FDA.
Yes they can. They can give you whatever medical advice or prescribe whatever regimen they want. They could tell you to drink bleach and the FDA would be powerless to do anything about it.
I think the strongest argument for you would be a drug approved in another country; but not in the US. Your doctor can still prescribe it; and under the right conditions that prescription can allow you to obtain the drug in a foreign country and bring it back to the US.
https://www.fda.gov/consumers/consumer-updates/5-tips-travel...