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A squat and priggish man of 46, Abdul Lateef Al Kindi has a thick salt-and-pepper beard and a reputation for causing controversy. During a sermon last August at his mosque in Srinagar—one of the capitals of Kashmir, and its largest city—he evoked the spirit of Islam as observed fourteen centuries ago, in the Prophet’s time, and demanded a total break from local traditions. He railed against the veneration of the tombs and relics of saints—common practice in Kashmir—as vestiges of ancient Greek and Hindu mythologies with no place in Islam.

Historically, Kashmir has been dominated by Sufi Islam, a mystical branch of the faith that the puritanically minded abhor. But Al Kindi plans to change all that. In a region already wracked by internal division and foreign...