I would propose that if one accepts the notion that there is a "christian nation" (which I do not, but I recognize that some nations have more Christians than others) then I would claim that this could be so, but that usually one side's soldiers would be guilty of sins, based on whether their side was morally right or not.
For instance, "christian" USA fought "christian" Germany in WWII. I would claim that German soldiers, unless they were completely without the news, would be guilty of aiding and abetting evil and that they could not have served in the German Army and especially the SS without moral stain, unless in some sort of hospital or chaplain capacity.
If the US were to engage in an unjust war, then christian soldiers would be forced to refuse to participate or else they would be committing the sin of aiding an unjust cause.
Ditto.
To simplify the issue, I think it is helpful to avoid the "Christian nation" issue entirely, and bring it to an individual level.
In that instance, the principle would be: Any time two Christians are on opposing sides in battle, at least one of them is in very grave sin.
Biblically unjustified killing is the definition of murder. And war is killing. Therefore, fighting in Biblically unjustified wars is murder.
At least one of the Christians has a Biblical duty to lay down his arms and refuse to 'serve his country.'
In today's sin-ravaged world, most times both sides are unjust, and neither Christian belongs on the battlefield.
I'm not sure if I diverted the thread too far from its original purpose, but I think it speaks to the same issue in an easier way. At least one side will always be unjust in a conflict. If the people as a whole in these "Christian nations" cannot rightly discern when to kill, then I think their status as a "Christian nation" is highly in doubt. So, no, Christian nations will never fight each other. Both may claim that status, but one (or both) will be wrong about their identity and the conflict itself.