Before the Greek War of Independence: the Orthodox Church was hostile to nationalism. Nationalism was this newfangled, godless French thing that set the people against their god-appointed ruler. The Patriarchate was particularly outspoken against it, and described it as a heresy.
That’s nationalism. Yes, you can say the Orthodox church helped preserve a notion of Greek identity. But that notion was safely ensconced within the Ottoman millet system, and the Orthodox church was a beneficiary of it: they certainly didn’t chafe against it.
The Greek War of Independence was formally started, as all Greek schoolchildren learn in school, by Germanos bishop of Old Patras on 25 March 1821. (They don’t learn that skirmishes had started a few days earlier.) And clergymen like Papaflessas and Athanasios Diakos were at the forefront of the fighting. But that was local lower clergy; it was not the policy of the church (though the Patriarch was hanged anyway).
Once Greek became an independent state, the Orthodox church was in an awkward position. It was still administered out of Istanbul, and its senior administration had obviously been an intrinsic part of how the Ottomans had ruled Greeks—so it had been collaborationist.
The former embarrassment was eventually rectified by the establishment of the autocephalous (“locally led”) Church of Greece, in 1870. The latter was rectified with some creative propaganda, such as the notion of the “secret school” (Krifo scholio)—that monks kept Greek culture alive through underground education. (No mainstream historian believes that to be true; after all, where did all the educated clergy come from, and how would the millet system have functioned without them? Few Greeks outside academia believe it to be false.)
Since independence, the Church has been firmly enmeshed in Greeks’ notion of what it is to be Greek: Orthodoxy is the state religion, Orthodox clergy preach loyalty to the state and open schools and parliament, the Greek church regards itself as the custodian of Greek national virtues.