President Barack Obama has authorized using air power to defend a new U.S.-backed fighting force in Syria if it is attacked by Syrian government forces or other groups, raising the risk of the American military coming into direct conflict with the regime of President Bashar al-Assad.

U.S. officials said the decision ended a monthslong debate over the role the American military should play in supporting its few allies on the battlefield in Syria. Administration officials had been deeply concerned that defending the Pentagon-backed...