As noted in the comments, this depends on what you mean by "Starting a war".
Technically speaking, a war requires a formal declaration by Congress (though specific forms are subject to debate). As such, the last one started by the USA was in WW2, with declarations in 1941 (Japan/Germany/Italy) and 1942 (Romania and other Axys countries); which makes the answer to be Franklin Delano Roosevelt was the last one to start a war, and every president since then did not.
More broadly speaking, if you count congressional authorization for the war, the last (and really only) 3 presidents since FDR who did not start a war were Lyndon B. Johnson, Richard M. Nixon and Gerald R. Ford. So Ford is the answer to your question.
Ford also differs from LBJ and Nixon in the fact that he can't be said to have actively prolonged Vietnam War (since his tenure started AFTER Paris peace accords of 1973). In contrast, Nixon continued the war despite Congress rescinding Gulf of Tonkin Resolution.