WASHINGTON—Secretary of State Mike Pompeo backed continued U.S. military support for Saudi Arabia’s war in Yemen over the objections of staff members after being warned that a cutoff could jeopardize $2 billion in weapons sales to America’s Gulf allies, according to a classified memo and people familiar with the decision.

The move has fueled rising outrage in Congress, where a bipartisan group of lawmakers is trying to cut off American military aid for Saudi Arabia and the United Arab Emirates in their three-year-old war...