Not for new drugs, they don’t.
BUT: FDA approved drugs constantly get prescribed for non-approved reasons (off label use). In fact, some drugs are MOST useful and most prescribed for conditions they were not originally intended for or approved for. And there’s nothing barring folks from gathering data on how that works out. Gabapentin rarely ever gets prescribed for its approved use anymore (seizure control). 99% of its scrips are written for off label diagnoses, and there’s a lot of studies done on it for said uses.
But even so, making hospitals do it for ivermectin or plaquenyl to pre-empt lawsuits or patient complaints remains a waste of time and resources.