Mr. Obama’s supporters and opponents alike say he has largely followed through on those promises. But many of his critics say he has overcorrected, further eroding the national-security architecture Washington built in the Mideast and Asia during the Cold War and allowing avenues for terrorist groups such as al Qaeda and competing states to fill the security vacuum.

“The U.S.’s regional order in the Middle East is in disarray,” said Emile El Hokayem, a Mideast expert at the International Institute for Strategic Studies, a London-based think tank.

The renewed instability in Iraq and Afghanistan in recent weeks is rekindling criticism that the White House hasn’t pushed hard enough to maintain a U.S. military presence in these countries.

Many Middle East leaders also have said Mr. Obama has been too reluctant to use force, which has emboldened terrorist groups and rogue states. They cite the president’s failure last year to follow through on a threat to strike Syrian President Bashar al-Assad’s regime after it allegedly used chemical weapons on its political opponents.