It's pretentious tosh from beginning to end. Is it anti-Catholic? Not particularly. But that's only because it's too busy being anti-life, anti-human race, and anti-everything else. There may have been some elusive hint of redemption in the movie, but it's so swathed in despair and disillusionment that it doesn't amount to very much.
The central character is not so much a good priest as a (supposedly) good man who happens to be a priest. The sincerity of his faith is never questioned, but he never shows the slightest inclination to come to the defence of his Church when it is being criticized. And it's criticized incessantly throughout the movie, although, to be fair, it's made pretty obvious that the Church is simply serving as a handy scapegoat or projecting screen for the other characters.
And, boy, do these people have demons! Everybody in Calvary seems to be homicidal, suicidal, perverted, despairing, abusive or masochistic. Worst of all, everybody is infinitely self-aware, ironic and jaded. Calvary is full of dialogue that is clever but incredibly tiresome-- nobody ever says anything obvious or straightforward or naive. At one point the atheist doctor concedes that he is a stock character, one which he describes as being "one part humanism and nine part gallows humour". That kind of thing.
What a waste of time and money. I should have gone to Spiderman instead. I think this might be the last time I ever go to see a film that threatens to be in any way arty. Maybe I am a soulless cretin, but I simply don't see human life as being the sewer of degradation that art-house directors seem to consider it.
ADDED THE NEXT DAY: I've been thinking about this movie since I saw it, and it strikes me that what really bothers me about it is the sheer waste. (And I'm not just talking about this movie, but others like it.) The cinema is such a marvellous venue. It can be a portal to such magical worlds. It can be a setting for the most sublime celebration of life, and of art. It can be, to use that clichéd but evocative phrase, a 'dream factory'. And anyone who goes to the cinema is making a commitment of time and money that may be very far from trivial to them.
And for all that investment to be thrown away upon an ugly howl of protest against the human condition seems to me almost blasphemous. Whether you believe that we are here because of Divine Providence, or whether you believe we are here because of a mind-bogglingly complex and unlikely set of circumstances, the outcome of a process that went on for millions of years before our birth, surely there's something sick in spending our time brooding on ugliness, evil and ennui?
Give me the cheerful humanism of Star Trek: The Next Generation over this kind of time-wasting, soul-wasting, money-wasting, life-wasting rubbish any day.