So you want to give into their premise? Do we also call this war a battle of civilizations?
Let's enumerate a bit.
There's a bunch of Sunni extremists, who subscribe to Sunni Islam, supported by former Baathists and remnants of Saddam's military. There's a bunch of young thugs in Somalia, another bunch in Mali and Nigeria. Then there's the Pashtun tribal group in Waziristan, more or less overlapping the Taliban. There's a bunch in Yemen. You might think of a few more.
But while your run-of-the-mill Trumpist might say this is a conflict between Islam and Christians, it's also grounded in Islamic factions against Israel and the Jews. When you enslave Kurds and Yazidis or Shiites, rail against "Western Education" for women, shoot up people at the theater and plot three years to kill your co-workers, it looks more like an assault on civilization.
But the end-game -- destroying civilization as we know it -- could only materialize if they somehow manage some cataclysm in such a way that it draws first-world superpowers into a mistaken conflict.
As for calling it a war against "The West," in that context and in a wider historical view, there is no major superpower or group of countries (the EU) that isn't part of "The West." China -- is now very much a "western" country, and China's transformation under communism was considered "the Triumph of the West" because Marxism and its communist offshoot is basically a western idea.
They don't manufacture their own cell-phones, laptops, Toyota trucks fitted with AA guns. But they need all that stuff -- artifacts of the civilized world.
If calling it a war against civilization makes them look as insane as they are, then call it that. I wouldn't give them the credit, though, of being an instigator of "World War III." I wouldn't even call it a "war," provided we can retake the land they've seized. I'd call it a police-action to kill, capture, incarcerate and even "medicate" the pathetic primitives that they are.
Above all else, they reject secularism and secular government. As someone said, they're deathly afraid of the modern world in those circumstances.