I'm not sure on this one, so I'm open to counter-examples.
First, note that there is a difference between necessary and sufficient. I'm suggesting that Christianity was a necessary, but not sufficient condition. Slavery was ended by British Evangelicals who made a tremendous political stink about the practice. That story can be seen in the movie Amazing Grace.
In any case, I saw this suggested in a Peter Kreeft book. Thinking about it, I couldn't think of a counter-example from a major league society. And, no, I don't want to hear about some Caribbean tribe of 140 people living in peace and harmony with nature. I'm talking nations on the order of Greece, Rome, Babylon, the Aztecs, the Ashanti and so forth. As far as I know, they all had slaves.
Going further, one might be able to suggest that given enough time, one of these groups would have eliminated slavery. Fair enough, but if you want to make that point, you need to show the first principles that would have led them to do so. Slaves are awfully convenient, particularly for those in power. Who wouldn't want a few concubines laying around the house to keep you amused when you get home from work as the CFO of Babylonian Rubber and Tire Company?
So go for it, folks. I'm not emotionally invested in this, so any argument in the comments can be completely collegial.