The two quotes that occured to me off the top of my head seem more overtly Christian. The first, what Scrooge's nephew had to say in inviting his uncle to dinner, "But I am sure I have always thought of Christmas time when it has come around--apart from the veneration due to its sacred name and origin, if anything belonging to it can be apart from that"
And then, when Bob returns from church he said of Tiny Tim, "that he hoped the people saw him in the church, because he was a cripple, and it might be pleasant to them to remember, upon Christmas Day, who made lame beggars walk and blind men see."
I think in the context of Victorian England, when virtually everyone was Church of England and that would be the background to any thoughts on morality and mortality, that you can't say it's not Christian, even if it doesn't seem overtly Christian to our more multi-cultural, areligious mindset.
That said, I think if all the Christian references were excised, it would be pretty much the same story, although not with the staying power that being related to Christmas gives it.