JERUSALEM—Unlike Syria’s other neighbors, Israel has by and large stayed away from the war that is ripping the country apart. But that doesn’t mean it isn’t deeply interested in how the five-year conflict ends—or that its interests are necessarily aligned with Washington’s.

The Israeli government’s priority is clear: to stop the rise of Iran as a regional power following last year’s nuclear deal and the lifting of international sanctions on Tehran. It is an approach that has increasingly aligned Israel with the anti-Iranian,...