Omar Akersim prays regularly and observes the dawn-to-dusk Ramadan fast. He is also openly gay. Akersim, 26, is part of a small but growing number of American Muslims challenging the long-standing interpretations of Islam that defined their parents’ world. They believe that one can be gay and Muslim; that the sexes can pray shoulder-to-shoulder; that females can preach and that Muslim women can marry outside the faith—and they point to Quran passages to back them up. The shift comes as young American Muslims work to reshape the faith they grew up with so it fits better with their complex, dual...