JERUSALEM — Under persistent prodding from President Obama, Israel and Turkey resolved a bitter three-year dispute on Friday with a diplomatic thaw that will help a fragile region confront Syria’s civil war, while handing the president a solid accomplishment as he closed out his visit to the Middle East.

The breakthrough took place in the most improbable of surroundings: a trailer parked on the tarmac of Ben-Gurion International Airport. Moments before Mr. Obama left for Jordan, Prime Minister Benjamin Netanyahu telephoned the Turkish prime minister, Recep Tayyip Erdogan, and apologized for deadly errors in Israel’s 2010 raid on a Turkish ship that was trying to bring aid to Palestinians in Gaza.

After years of angrily demanding an apology, Mr. Erdogan accepted Mr. Netanyahu’s gesture, and both sides agreed to dispatch envoys to each other’s nations, having recalled them in 2011....