How things have changed since the late 1950s! On the one hand, we no longer accept the inevitability of the destruction of civilisation by nuclear war (which Miller has happening not once but twice). The Cold War seemed inevitable and unescapable in 1959, and indeed for most of the next three decades. We have different concerns now.
On the other hand, it's difficult to imagine a serious writer today taking such a positive view of the Church. Benedict XVI is very different from John XXIII; the Vatican has boxed itself in politically. There is a greater tension now than then between religion and science, thanks to the foolishness of the religious right on the one side and the determination of Richard Dawkins to miss the point on the other; Miller's understanding of the Church's role in the Dark Ages has itself been weakened (though not totally disproved) by later scholarship. And Miller is able to largely ignore sex and women in his novel, which I think would be impossible for anyone writing about the Church today.
For all that, it's a great book, and rightly won the Hugo in 1961 (of its rivals, I've read Budrys' Rogue Moon and Anderson's The High Crusade) Miller explores faith, history, tradition, political engagement, the advancement of science, and grace; and does it all with a wry and sympathetic humour. The Wandering Jew who pops up from time to time in the narrative is thought by some romantics to have been inspired by Judith Merrill, but clearly comes from many sources (and is occasionally the author's own viewpoint). And there is the continuing enigma of Mrs Grales at the end.
It is sad that this was basically the end of Miller's literary career. He seems to have agonised for thirty-five years over Saint Leibowitz and the Wild Horse Woman before ending his own life, and never published another story.