Bear with me, as there will be much convergence and handwaving in this post, but there is a point to it.
Say that world history generally and Anatolian/Balkan history specifically goes roughly the same as in our world. A branch of Oghuz Turks defeat the Byzantine Empire in a counterpart to the Battle of Manzikert in the 11th century, opening up Anatolia to Turkish settlement, and one of the successor states to these initial invaders establishes an empire in Asia Minor and the Balkans that eventually captures Constantinople... without converting to Islam. Perhaps the Seljuq leadership does not convert to Islam in the 10th century (though Wikipedia seems vague as to whether they were Christian, Jewish, Tengrist, or something else beforehand). Once a group of them are firmly established in Anatolia, they opt to convert to Nicene Christianity (though their cousins to the east will probably still become Muslim, if somewhat later than in our world). Upon taking Constantinople, the counterpart to Suleiman the Magnificant claims the title of Roman Emperor, but the Turks otherwise remain culturally distinct from the Greeks.
This is a great deal of build-up and convergence to ask how the rest of Europe would interact with what is essentially a Christian version of the Ottoman Empire. I assume that those in Western Europe would take this state's claim to be the Roman Empire about as seriously as they did the Greek claim, but they would have to acknowledge that this new empire's strength sets them apart from the ineffectual Byzantine Empire. These Ottoman counterparts would still likely be political rivals of various Italian city-states, the Austrians, the Hungarians, the Russians, and other nearby countries, but they would not be considered an existential threat to Christendom, either. How do events unfold?
Say that world history generally and Anatolian/Balkan history specifically goes roughly the same as in our world. A branch of Oghuz Turks defeat the Byzantine Empire in a counterpart to the Battle of Manzikert in the 11th century, opening up Anatolia to Turkish settlement, and one of the successor states to these initial invaders establishes an empire in Asia Minor and the Balkans that eventually captures Constantinople... without converting to Islam. Perhaps the Seljuq leadership does not convert to Islam in the 10th century (though Wikipedia seems vague as to whether they were Christian, Jewish, Tengrist, or something else beforehand). Once a group of them are firmly established in Anatolia, they opt to convert to Nicene Christianity (though their cousins to the east will probably still become Muslim, if somewhat later than in our world). Upon taking Constantinople, the counterpart to Suleiman the Magnificant claims the title of Roman Emperor, but the Turks otherwise remain culturally distinct from the Greeks.
This is a great deal of build-up and convergence to ask how the rest of Europe would interact with what is essentially a Christian version of the Ottoman Empire. I assume that those in Western Europe would take this state's claim to be the Roman Empire about as seriously as they did the Greek claim, but they would have to acknowledge that this new empire's strength sets them apart from the ineffectual Byzantine Empire. These Ottoman counterparts would still likely be political rivals of various Italian city-states, the Austrians, the Hungarians, the Russians, and other nearby countries, but they would not be considered an existential threat to Christendom, either. How do events unfold?