No one would frame it in religious terms with the possible exception of Russia, but even for them, I'm not how sure how big a role religion played when they conquered the Muslims in the Caucasus. Thus, it wouldn't so much be a crusade as much as a blatantly imperialist dismantling of the Ottoman Empire. You can tell because ultrareligious Bourbon Restoration-era France didn't declare it a crusade when they attacked Algeria, for instance.
I think the Muslims would be the ones to seize on the religious war angle long before the Christians.
That said, any attempt at "crusading" during the 19th century would probably be a victory for the Christians, it's just as the Crimean War showed, they'd be just as much having to fight other Christians in the name of the balance of power in Europe as they would Muslims. Because any attack against the Ottoman Empire was considered as helping Russian expansionism out.
Further, the "crusader states" would find themselves forced toward secularism (except for maybe one in the Lebanon area), since that would be the only way to get the majority Muslim population on their side. And Muslims wouldn't necessarily mind, if the popularity of secularist nationalism throughout the 20th century shows. Remember that groups like the Syrian Social Nationalist Party and the Ba'ath Party both prominently had Christians in their movement like Michel Aflaq and Antun Saadeh. But all of them knew you had to have the Muslims on your side in order to take power and create nationalism.