Indian Prime Minister Narendra Modi didn’t only lay the cornerstone of a new Hindu temple last week. He was also setting the foundation for a new order in the Indo-Pacific: one in which India will have a closer relationship with countries like the U.S., Japan and Australia, but in which Hindu-Muslim and India-Pakistan tensions will complicate efforts in Washington and Beijing to keep U.S.-China relations from spiraling out of control.

Mr. Modi’s many critics in India and the West have focused on the domestic consequences…