SHANGHAI—This was the humiliating result that China feared most: A small country, the Philippines, took on and comprehensively defeated the aspiring regional hegemon in an international court of law. Beijing can’t back down.

In the aftermath of an unambiguous and unanimous legal verdict that strikes down Beijing’s historic claims to the South China Sea, rebukes it for turning coral reefs into island fortresses through massive dredging and sides with bullied Philippine fishermen, the Chinese government’s legitimacy is on the...