Yes. The diplomatic term for this to declare someone persona non grata. It was a relatively common procedure in the Cold War, particularly for individuals suspected of using diplomatic cover for espionage, or officials perceived of breaching diplomatic protocols. The ability to do so is currently defined under the Vienna Convention on Diplomatic Relations which simply says that "the state" may make such a declaration, so in a sense it need not literally be the president, but in a US context the State Department is part of the executive branch and ultimately derives its authority from the President.