BEIRUT—Iran aired footage Tuesday of what it said was a U.S. drone it had captured over Iranian airspace in the Persian Gulf, a claim the U.S. Navy denied.

The incident underscored the tensions, and behind-the-scenes surveillance battles, between the West and Iran over Iran's nuclear program. Iran has targeted U.S. drones before and used its announcement to burnish its defense credentials. Americans familiar with the defense industry, meanwhile, underplayed the intelligence significance of the craft Iran said it had captured,...