JEDDAH, Saudi Arabia—For Saudi Crown Prince Mohammed bin Salman, his high-profile meeting with President Biden was an immediate victory, signaling that his isolation on the world stage is over. But the American president will have to wait weeks, perhaps months, before the world knows if his sojourn to the Middle East paid off for the U.S. On Saturday, Mr. Biden left Saudi Arabia, the second and final stop on a four-day Middle East tour that also took him to Israel. His aim was to reaffirm U.S. ties with an oil-rich region, following a rift with Saudi leaders after Mr. Biden...