John Piper says the following on this issue:
I sympathize with those who want to be rigorously and distinctly Christian, who want to be disentangled from the world and any pagan roots that might lie beneath our celebration of Christmas, but I don't go that route on this matter because I think there comes a point where the roots are so far gone that the present meaning doesn't carry the pagan connotation anymore. I'm more concerned about a new paganism that gets layered on top of Christian holidays.
Here's the example I use: All language has roots somewhere. Most of our days of the week—if not all—grew out of pagan names too. So should we stop using the word "Sunday" because it may have related to the worship of the sun once upon a time? In modern English "Sunday" doesn't carry that connotation, and that's the very nature of language. In a sense, holidays are like chronological language.
Christmas now means that we mark, in Christian ways, the birth of Jesus Christ. I think the birth, death and resurrection of Christ are the most important events in human history. Not to mark them in some way, by way of special celebration, would be folly it seems to me.
I remember I lived next door to somebody back in seminary who didn't celebrate birthdays for their kid. The idea was, partly, that all days were special for their kid. But if all days are special then it probably means that there are no special days. Yet some things are so good and precious—like anniversaries, birthdays, and even deaths—that they are worthy of being marked. How much more the birth and death of Jesus Christ!
It's really worth the risk, even if the date of December 25 was chosen because of its proximity to some kind of pagan festival. Let's just take it, sanctify it, and make the most of it, because Christ is worthy of being celebrated in his birth. (Source: Desiring God)
I also sympathise with those who want to keep Christianity distinct from the things of the world. Nevertheless, whatever pagan connotations may have once existed are now no longer salient. For those who argue there are no special days in the Christian calendar, as Piper rightly points out, some events are so momentous they deserve to be marked out as such. The view there are no special days for the Christian is made more bizarre when it is held in conjunction with a strict sabbatarianism that states Sunday really is special for the Christian. So, on the one hand, Sunday is special (and it is often argued that the Sabbath moved from Saturday to Sunday because this is the day Christ was resurrected - an event special enough to be remembered) but, on the other, Christmas is not.
At the heart of the matter, the birth and death of Jesus Christ are the two most important episodes documented in scripture. To not celebrate these events does little to indicate Christians are distinct and separate from the world and far more to suggest we don't think these events matter - which of course, they do! So, let's celebrate Christmas as a means of showing the world that the birth of Jesus Christ really is important and really is something worth celebrating.
