UNITED NATIONS—President Barack Obama outlined a U.S. approach to the Middle East for his last two years in office that tightly focuses on diminishing Islamic extremism and leans heavily on American military power, as he implored regional leaders to do more to combat what he called the single most pressing threat to global progress.

In his sixth address to the United Nations General Assembly on Wednesday, Mr. Obama made clear that fighting what he called “the cancer of violent extremism,” embodied in groups such as Islamic...