Over seven decades, the United States and Saudi Arabia forged a strategic alliance that became a linchpin of the regional order: a liberal democracy and an ultraconservative monarchy united by shared interests in the stability of the Middle East and the continued flow of oil. But with President Obama arriving in Riyadh on Friday, the rulers of Saudi Arabia say they feel increasingly compelled to go their own way, pursuing starkly different strategies from Washington in dealing with Iran, Syria, Egypt and the role of the Muslim Brotherhood in the region. “Their view of Mr. Obama is that his entire...