PARIS — Tensions between Saudi Arabia and the United States over Washington’s approach to the Middle East were brewing for months before they burst into the open last week. First, there was the American inaction in Syria and lack of progress on Israeli-Palestinian peace. Then came America’s withdrawal of aid to the Egyptian military after the July coup. Now President Obama is pursuing a very public rapprochement with Iran, Saudi Arabia’s archrival. The mounting disagreements between the two longtime allies is now in full public view. Last week, the head of Saudi intelligence warned that it would stop cooperating with...