PHILADELPHIA — Cardinal Gerhard Ludwig Müller, the Vatican’s chief doctrinal watchdog, sat cross-legged in front of a large television on Friday and watched Pope Francis at the United Nations addressing a world that has warmed to the pontiff’s merciful message on issues like immigration, the poor and climate change.

“If the pope speaks about social justice, everybody will embrace him, no?” the cardinal said. “It’s not so difficult. But to speak about moral values, and the field of sexuality and matrimony and abortion and these values, is more conflictive.”

The cardinal, a conservative German in black clerical clothing, said neither the pontiff, nor his church, cared whether “Obama says the pope is a very good man” or whether a “fallible” Supreme Court legalized gay marriage. And if papal proclamations of Catholic doctrine on core issues of family have eroded Francis’ global standing, so be it.

But Cardinal Müller is no objective papal observer. He is a leading voice in the orthodox wing of the Catholic Church that worries that outsize attention on Francis’ welcoming, pastoral style could distract from the church’s core beliefs.