ROME—When Pope Francis said he wouldn't judge gay priests, he opened the door to a new era of reconciliation within the Roman Catholic Church, which has struggled for decades to confront the presence of homosexuality in its ministry.

The pontiff was traveling aboard a turbulent overnight flight to Rome from his first overseas trip—a journey marked by his plain-spoken appeals to Catholics to reground the church in grass-roots ministry—when he broached the delicate issue of how the Catholic hierarchy should respond to clerics...