Final answer:
John Adams was the President of the United States when the US Navy Department was established. He was elected in 1796 and his vice president was Thomas Jefferson.
Step-by-step explanation:
The US Navy Department was established during the presidency of John Adams, who was the President of the United States following George Washington. John Adams, a Federalist, won the election of 1796 and, interestingly, his Vice President was Thomas Jefferson, a Republican, due to the electoral system in place at the time. This occurred during a period when the new nation was defining its stance on foreign policy and its internal governance structure.