Can doctors write prescriptions without seeing patients?

No and yes. A doctor can only write prescriptions if he or she is physically in the state in which he is licensed. If the doctor is calling in a prescription to a pharmacy, obviously the pharmacy wouldn't know what state the doctor was in, so this is what most doctors do if travelling outside of the state they are licensed in. However, a doctor can't practice in a state that he or she is not licensed in, so they cannot physically write a prescription while in another state.