I hope I am conveying this in the right sense. My point is that war only aids the extremist cause, it does not hurt it. If anything it makes it more powerful.
I get what you're saying. Let me just criticize a few things. You're not entirely wrong but you're not entirely right either.
But a war between Christianity and Islam wouldn't be based on casualties or deaths.
But it would for one simple reason. If Europe and the Americas (the "christian" nations) completely Conquered the Middle East. It wouldn't matter if they ideologically felt they won because they're martyrs. They lost the war. they lost their men and boys. They would then have a middle east which was rebuilt in a European image, similar to how the United States rebuilt Japan after WW2.
If the two religions were at war tomorrow, Islam would win. The Christian world has become secularized and far less extreme then it once was.
While this is true, people tend to return to their religions when they feel their nation is at threat. You're also ignoring certain countries that while they're not as powerful as the US, they're still very strong and very Christian.
Russia and all of Eastern Europe. They are VERY Christian. To be Christian (Orthodox or Catholic usually) is to be that nation. To be Greek is to be an Orthodox Christian. To be a Croat is to be a Catholic. They're inseparable in most people's minds. So if they felt Christianity was at risk they would not hesitate to fight back in the name of Christianity.
Then you take into account the United States, arguably the most powerful nation on earth, has some of the highest rates of religious devotion in it's population. The United States is also the prime target in this. So the US would also fight.
That's if we're talking about Day 1 being peace and Day 2 being war. Meaning an instant change, these are the nations which would have a vested interest in "defending the faith".
If we're talking about an attack on "Westernism" which is what we are primarily (Christian nations really just means non-muslim nations in Europe) the Entire European continent, Russia, Australia, The Americas (including south america) and Israel, would fight AGAINST such powerful countries as Turkey, Saudi Arabia, Syria, Lebanon. These are not countries which could stand up to the United States, let alone the entirety of the rest of the world.