When Pope Francis delivers his Christmas message this weekend, he will do so not just as the head of the Catholic Church but as the improbable standard-bearer for many progressives around the world.

With conservative and nationalist forces on the rise in many places and with figures such as U.S. President Barack Obama and French President François Hollande on their way out, many on the left—from socialists in Latin America to environmentalists in Europe—are looking to the 80-year-old pontiff for leadership.

“Pope...