THE last time the Chair of St. Peter stood vacant, during Pope John Paul II’s funeral in 2005, the Roman Catholic Church enjoyed a wave of unusually favorable coverage from the American press. The Polish pope had a way of disarming even his most stringent critics, and that power extended beyond his death, turning his funeral into a made-for-television spectacle that almost felt like an infomercial for the Catholic faith.

Perhaps not coincidentally, the mid-2000s were the last time the Catholic vision of the good society — more egalitarian than American conservatism and more moralistic than American liberalism — enjoyed real influence in U.S. politics. At the time of John Paul’s death, the Republican Party’s agenda was still stamped by...