RIYADH, Saudi Arabia—Secretary of State John Kerry and his European allies, in their renewed drive to promote stability in the Middle East, are increasingly reliant on the two countries the U.S. has historically argued only undermine stability—Russia and Iran.

The engagement of Moscow and Tehran by the U.S. and Europe exemplifies the shifting balance of power in the region, as President Barack Obama has made clear his intent to minimize the U.S. military role there, according to U.S. and Arab officials.

But...