The fun question here is "what is a christian nation?" and "would they have done it given the chance?" There are many intolerant butchers throughout history who simply lacked the power and assembly line mentality of Hitler and Stalin.
The questions are complicated, IE is Hitler a christian? My answer, for practical argument (my definition being: One who professes to believe in Jesus Christ the son of The Living God, who came to earth and died for the sins of men and rose from the dead. I use this definition purely for cultural argument, I'm a christian and I believe quiet firmly he will go to hell.), is yes. Were his actions strongly influenced by the Church as founded? Somewhat, its not a sufficient cause.
How would you come to the conclusion that Hitler was Christianm? I camn't recall any instance of him ever professiong faith during his politzical career - respect for the Church and high regard for Christian culture, yes, but faith not.
then again, as was pointed out before, you can't grow up in a Christian society and not pick up some Christian values, so the point is a bit moot.
Was the church complicit (and this is the important part)? Maybe. Maybe not.
It all depends on when the Church learned of it. According to some readings, the Catholic Church knew well before the allies, whereas according to other sources the Catholic Church didn't know until it was announced by the allies and the Catholic Church doesn't need to have a papal statement to condemn mass murder.
It is not just about the Shoah. Many Christians in Germany and elsewqhere stood up against Nazi dictatorship, and they did so in direct contravention of the word of the established churches. Catholic and Protestant churches signed concordates with Hitler early on (you could argue a degree of ignorance) and stood by them (which puts the kibosh on that defense). Church archives were opened to registry officials in 1935 for the purpose of defining whose ancestors were baptised Jews. Church-owned businesses and charities used convict and slave labour. Church authorities reprimanded, sanctioned and fired people who spoke up against the system. If that isn't complicit, it is hard to say what degree of cooperation would be. Neither church operated death camps.
Lets take the Inquisition for instance, that looks like a really evil thing, church sanctioned. And indeed it was both, but here's the issue, you know why the Inquisition in France killed a handful of people and the Inquisition in Spain killed hundreds? Because the government pressured them, because it pushed and controlled them. The Inquisition was just a Gulag with a religious name stuck on and a less efficient application. Would the inquisition have been kinder or colder without the Church? I do not know, maybe they would just be burned or flayed or tortured without any chance of getting out of it. Maybe the inquisition would have just chopped off some heads. I certainly don't know.
Again, you can't be born in a Christian country without picking up Christian values - that goes for the kings of Spain as much as for modern secularists. The Inquisition is more than a secret police because of its remit. Rome's frumentarii and delatores were as paranoid and ruthless as they come, torturing and murdering whoever they thought a danger to the regime. Many dynasties in China ran similar operations. The idea is so common that even the Zulu are supposed to have had something like it. The difference was, though, that these agents were focused on individuals who committed certain *acts* (even if that act was no more than an incipient conspiracy). This could go as far as the worst excesses of mass paranoia do, though these instances are rare and temporary. The Inquisition (it shares this distinction with many organisations in the world after the October Revolution, both Communist and non-Communist) prosecuted people for beliefs that they held. This was not incidental, it was central (and the reason why they handed witches and active heretics over to the secular arm - they had committed criminal acts, so the fathers washed their hands of them). An institution that has free rein in prosecuting thoughtcrime is a relatively new invention that you can only get if you marry religious orthodoxy to state power - invented (probably independently) in imperial Rome, Liang China, Sassanid Persia and the Muslim world.
The question of whether or not Christianity is a positive influence is absurd because there is no way to prove it, one way or another. The only way to prove it would be to see a world without it. And, as we all understand here at Alternate History Dot Com, there's always butterflies.
It would always be possible to point to South and East Asia as a quasi-experiment. As far as I can see they have broadly the same level of nastiness, but less systematic internal savagery (until they catch Westernitis) and about the same level of personal morality based on roughly the same precepts.