As Fred reminded us in a comment, 8 USC § 1185(b) (Immigration and Nationality Act of 1952, § 215(b)), “Travel Control of Aliens and Citizens”, demands that:
Except as otherwise provided by the President and subject to such limitations and exceptions as the President may authorize and prescribe, it shall be unlawful for any citizen of the United States to depart from or enter, or attempt to depart from or enter, the United States unless he bears a valid United States passport.
This requirement traces its roots all the way back to the 1952 INA as first enacted. However, it’s worth remembering two rather amusing facts about the law as it originally stood: it only required United States citizens to bear a “valid passport” and not a “valid United States passport”, and it only applied in time of war or national emergency:
(a) When the United States is at war or during the existence of any national emergency proclaimed by the President, or, as to aliens, whenever there exists a state of war between or among two or more states, and the President shall find that the interests of the United States require that restrictions and prohibitions in addition to those provided otherwise than by this section be imposed upon the departure of persons from and their entry into the United States, and shall make public proclamation thereof, it shall, until otherwise ordered by the President or the Congress, be unlawful …
(b) After such proclamation as is provided for in subsection (a) has been made and published and while such proclamation is in force, it shall, except as otherwise provided by the President, and subject to such limitations and exceptions as the President may authorize and prescribe, be unlawful for any citizen of the United States to depart from or enter, or attempt to depart from or enter, the United States unless he bears a valid passport.
So when did these two facts change, you might ask?