In the years since the U.S. invasion of Iraq in 2003, American promises of turning the country into a model democracy, spreading freedom across the Middle East, have often seemed like cruel mockery. By the time the U.S. withdrew in 2011, Iraq had been ravaged by bloody insurgencies and sectarian massacres that killed hundreds of thousands of Iraqis and more than 4,500 American troops. In 2014, Iraq almost collapsed in the face of a blitzkrieg by Islamic State, as the extremist group reached the outskirts of Baghdad. There weren’t many takers in the region for the Iraqi model.

Today...