Is it correct that only Orthodoxy kept the Greek language alive? Were non-Christian Greeks not speaking Greek up to the 1900s?

It’s only correct that Orthodoxy kept the Greek alphabet alive; scripts in the Ottoman Empire were associated with creed. Thus, according to the creed of the Greek speaker, Greek was written in

• Greek script (Orthodox),
• Latin script (Catholic: the Franco-Levantines, including many works of the Cretan Renaissance, and in the Aegean sponsored by Jesuit schools),
• Hebrew (Jews: the Judaeo-Greek Pentateuch of 1547, and some other songs and religious texts in Judaeo-Greek), and
• Arabic (Aljamiado literature, written by and for Muslim Greeks).

Just as Turkish was written in Greek script for Turkish-speaking Christians (Karamanlides). There would be no Aljamiado literature, of course, if Greek Muslims didn’t want to write and read in Greek.

Aljamiado literature (a term borrowed from its Spanish counterpart) has been ignored by Greek scholars until recently. A 2014 lecture on the literature is available at Τουρκογιαννιώτικα στιχοπλάκια και τουρκοκρητικές μαντινάδες: Η ελληνική aljamiado γραμματεία (inaudible Turkish and Greek, with audible but halting simultaneous translation into English).

EDIT: Btw, the notion you will occasionally hear in Greece, that the Greek church somehow preserved Greek is actually bundled up in the notion that the Greek church preserved Greek identity and Greek learning during Ottoman Rule. The Rum Millet, you can argue, did in fact do so; but that was not a Greek idea, but Mehmed II’s. (Dimitris Almyrantis, I’m fishing for an answer from you here.)