ROME—Reports that Pope Francis told a sex-abuse victim that God made him gay have drawn headlines this week, with many observers inferring a new level of acceptance of homosexuality, which the catechism of the Catholic Church describes as “objectively disordered.”

The Vatican spokesman declined to confirm or deny the statement, citing a policy of not commenting on the pope’s private conversations.

But no aspect of Pope Francis’ five-year reign has been better known or more controversial than his conciliatory approach to gay people, most famously expressed in his 2013 words about gay priests: “Who am I to judge?”

Conservatives have been prompt to argue that Pope Francis’ statements on homosexuality, confirmed or otherwise, don’t conflict with church teaching or suggest toleration of homosexual acts, but simply reflect the basic Christian message of God’s love for all people.

Yet the pope’s stance has encouraged a greater openness to homosexuality among the church’s hierarchy, leading to calls for the appreciation not just of gay people but of gay relationships. That trend is straining any appearance of a consensus on the subject within the church.