He might not look it, but Bishop Robert Barron is a social-media superstar. Sporting thick-framed glasses and a folksy Midwestern demeanor, the Catholic auxiliary bishop of Los Angeles resembles the comedian Drew Carey, only trimmer. But the aw-shucks routine isn’t fooling anyone. Beneath it lies a penetrating intellect and clear talent for using popular culture to draw people into—or back to—Catholicism.

With...