Pope Francis arrived in Portugal on Friday to commemorate the 100-year anniversary of reported apparitions of the Virgin Mary at Fatima—honoring the sort of grass-roots devotion that can bolster the Catholic faith but also pose challenges for the Vatican.

Apparitions and similar experiences can spur deep piety among believers yet draw skepticism or even ridicule from Catholics and non-Catholics alike. The church accepts the apparitions at Fatima as what it terms “worthy of belief,” and it is examining several dozen similar...