What is the "modern era" in this context? This being pre-1900, I'm assuming something like "after the Enlightenment" is meant-- or do you mean the broadest interpretation of "modern", which includes
basically everything as of 1500 or so? I'm asking because it really matters. For Christianity in North Africa and the Near East, "the sooner the better" is the adage by which a POD should be selected if one desires the greatest success. The history of Christianity in the Islamic World has been one of decline, and later PODs thus give it a greater disadvantage.
If we're talking about a fin-de-siècle POD here, then the map provided by
@RandomWriterGuy gives you the answers, really. In addition to OTL's creation of Georgia and Armenia, there would be potential for a much bigger Armenia, and Greater Greece, and a Christian state(let) in Lebanon, as well as maybe an Assyrian statelet. Cyprus could be entirely Greek and Christian as well. All this would require at least a degree of ethnic cleansing, although it could be limited to the population exchange variety (as opposed to the "mass graves" variety). None of this is happening without European involvement, just as the more modest developments of OTL didn't happen by themselves.
The more willing you are to entertain harsh methods, the greater the potential areas in question become. For instance, the above scenario suggests that an Armenia may be larger than OTL, but it certainly wouldn't be likely to include everything claimed by Armenian irridentists following OTL's Great War. To get such vast claims awarded (and cleared of mot or all Muslims, so as to ensure a Christian majority), you need to prepare for a very bloody scenario indeed. Still, there are things along those lines that could happen. Forinstance, given the right (or "right") premise, one could see the Islamic-majority areas North and North-West of Dagestan getting ethnically and religiously cleansed by a vengeful Russia-- thus making the region Christian by default. But given a POD this late, something like a viable Coptic state seems utterly unworkable, unless you're willing to commit full-scale genocide against the Muslim population. (On the other hand, the ruthless and brutal appoach could lead to, ah, "results" in Algeria-- just have the French plans to carve out a pied-noir state actually go through.)
So there are possibilities, even with a late POD, but they are limited, and they all require force. And the more expansive your goals, the more force you'll need to apply.
The further back you go, the higher the relative number of Christians will be, and the less brutal force you'd have to apply in order to establish Christian-majority regions. (The war to gain control of the region would still be brutal, of course.) If, for instance, a stable and strong anti-Ottoman coalition had formed somewhere in the 18th century, that could have seen the Ottoman empire carved up-- with several Christian states coming into existence as a part of that process. The percentage of Christians in all the aforementioned regions simply gets higher the further back you go. And if you're willing to go back to 1500 or thereabouts, the odds are even better. (Plus, of course, the fact that over five centuries, you can basically force Christianisation of any captured region by making conversion the
only avenue for upward mobility.)