Some that come to mind

The Byzantine iconoclastic controversy 8-9 centuries weakened the Empire.

Some bits of the ECW (bishops wars) where religious.

The northern Crusades (much more savage that the eastern ones)

The Hussite Wars

I think the modern version is Islamic- you see many Sunni nations more willing to work with Israel than Shia Iran.

Christianity is not an explicit part of any modern state outside of the Vatican that I'm aware of (with a small technical exception for Britain's monarchy.)

Mount Athos has special autonomous status within the Greek Republic; the Orthodox equivalent of Vatican City for the Catholic world.

(Not the OP)

How about the Albigensian crusade?

https://en.wikipedia.org/wiki/Albigensian_Crusade

Also, possibly, sectarian violence in Ireland.

Ireland's feuds aren't sectarian, they're ethno-nationalist. We call them "Protestants" and "Catholics," but really what those names mean are "the group in Ulster and especially in Belfast, whose ancestors colonized the land for England, and a few who cooperated with them" and "the rest of the country whose ancestors lost their war and had to put up with being second class citizens oppressed by the British military and police well into the twentieth century."

And even if the British monarch is a figurehead today, the legacy of violence goes back to the Tudors.