There are probably lots of very straightforward possibilities to make this happen. But given that it's a challenge that's a bit out there anyway, let's go for something cool. Here's a POD post 1500: the siege of Vienna in 1529 succeeds. Let's for the moment assume that as a given (maybe the campaign goes somewhat differently, and the Ottomans get to the city before OTL's heavy rainfall and then sudden unseasonable snowfall really messes with their plans). The forces of Suleiman the Magnificent now pose a danger considerable parts of Europe. The world of Christendom is, at the very least, utterly shocked. Let's assume that Suleiman, being his own highly gifted self, manages to make serious territorial gains. At the end of the day, his campaign leaves him with his OTL gains, as well as OTL's Royal Hungary, Austria proper, Styria, Carniola, Slavonia and Croatia. (I think Suleiman could potentially push for even more than that, but this would leave him vulnerable to a counterattack from other European powers; I imagine he would take the aforementioned gains and then consolidate, rather than risk it all on a wild gambit.)
This outcome means that the Ottomans now directly threaten Venice, Tyrol, Bavaria, Bohemia and Moravia. And potentially many regions beyond. OTL's failed siege put a stop to Ottoman ambitions in Europe, for a time. It preserved, in many ways, some semblance of the status quo. But this kind of ATL scenario? This conquest into the "heart of Christendom"? This would demand an answer. And there, I believe, is an impetus for an eventualy counterattack. A new "reconquista", followed by... well... a new crusade, of sorts.
At any rate, this ATL would have the Ottoman's on everyone's minds as enemy number one. Culturally, intellectually and socially, the Christian-Muslim conflict would be played up far more than it was in OTL. There might be more of a Christian sense of unity against this feared external enemy. There is of course no way to predict how this would turn out. But since this is a challenge, let's make it interesting. Let's say that this outcome leads to a rather broad alliance between at least France, Spain, Portugal, the Italian states, Bavaria, and what remains of the Austrian Habsburgs' hereditary lands (basically Tyrol, Bohemia-Moravia and Silesia). Nominally, the HRE would also be part of this alliance, probably. Since this is also a time of internal religious struggle, however, and since the Protestant regions are generally situated further away from the Ottomans, I wouldn't count on much involvement from any Protestant region. (On the flipside, a common external enemy might make both Catholics and Protestants somewhat more ready to tolerate each other, so the hated muslim foe cannot prevail by exploiting their division.)
The purpose of these Christian powers would be to push back the Ottomans, to liberate south-eastern Europe from the muslim yoke, and ideally to give the hated enemy a sound kicking and... take back the holy land or something. hat last bit is probably more propaganda than something else, but... a firm alliance of major Christian powers, at the exact time those powers are getting really into colonialism and the wealth that entails, has a fairly good chance of succeeding. So they push back the Ottomans. Through a long and hard struggle, that lasts (intermittently) for over a century, and that only increases the hate they feel towards these conquerors, they eventually push the Ottomans entirely out of Europe. They take Constantinople. By that point, the whole thing has taken on a religious dimension; a life of its own.
And thus, it becomes almost inevitable that, rather than marching into Anatolia proper, the next step is the launch a campaign to take the Holy Land. Many call it a new crusade. In many ways, it is. And this, too, is a success. (Let us, for the sake of flair, assume that the city of Jerusalem is captured on September twelfth, 1683.) A considerable part of the Levant is captured. The Kingdom of Jerusalem re-founded. In this bitter war, the outcome of the campaign includes the mass expulsion of muslims from the (re?)conquered area. In the following years and decades, Christians settle there. All that happens even as this victory proves to be the alliance's last one. Internal animosity, colonial conflicts and various other factors (such as always arise) lead the alliance to fall apart or at least to lose its sense of purpose. Maybe even that fact that the object of all crusades is now achieved, and that the war to get there was so long and costly, plays a role. In any case, the Ottomans are much diminished. They cannot take back Jerusalem, or any part of Europe they previously held. But the European powers, at odds in various constellations and coalitions once more, cannot gather the will to destroy the Ottomans completely, either.
And so, the new status quo is preserved. The Kingdom of Jerusalem remains, until the present day, a part of Christendom, and culturally European in almost every way.