BAGHDAD—The highest military-ranking officer in the U.S. on Monday ruled out immediate American airstrikes to stem Islamic State’s destruction of Iraqi archaeological and cultural treasures.

In the last 10 days, Islamic State militants have been blamed for at least three incidents in which artifacts and ruins dating back more than 3,000 years have been looted and destroyed, shocking Iraqis and provoking a chorus of international condemnation.

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