I have considerable respect, and a certain degree of (sentimental, I think) fondness for the One Holy Catholic and Apostolic Church. The fondness is in the nature of nostalgia as it is for the Church that was, which is to say more accurately the Church as I first knew it; the Latin Church, literally and figuratively.
Sancta Mater Ecclesiae. I knew it for a time even after that, when guitar masses were (I suppose I must say it) celebrated. I've attended a mass now and then since that time, for weddings and funerals. There's no beauty left. It's relentlessly prosaic.
That's the Church I think you refer to, even in its Anglican form. The Anglican Church retained much of the Latin Church despite the fact it came about largely because of the monstrous Henry VIII.
But that Church was a kind of mish mash, or hodgepodge, of ancient Rome and through it ancient Greece (though the Orthodox Church is probably truer to the Church as it morphed in the Eastern Empire, which lived on until finally destroyed by the Latin West). What took place in the Church intellectually in 11th and 12th centuries was primarily the result of the rediscovery of Aristotle. What took place politically after the Western Empire is traditionally said to have fallen, was a continuation of the Empire in many respects in the form of Gothic and Vandal successor Roman states, which sometimes squabbled with, sometimes cooperated with, the Eastern Empire, and was even reclaimed by the East for a time during Justinian's principate. Then Charlemagne was crowned Emperor, and The Holy Roman Empire gradually took form, and actually lasted in progressively diminished form until 1806. The Renaissance was inspired by the works of the ancient pagans. Thus, for example, Dante chose Virgil as his guide, and populated his first level of Hell with the great pagan thinkers living in comfort and discussing great things, though necessarily existing apart from the Christian God. The Church, as others have noted, is a kind of ghost of the later Roman Empire in its organization, its ceremony, and even its vestments, and I don't doubt it contributed to the stability of Europe.
Tertullian, who may have been the son of a centurion and a lawyer, was raised as a Roman in the Roman province of Africa. He's the one who said "What has Athens to do with Jerusalem?" Perhaps not much, but he was very much a Roman as were all the Church Fathers. He was just a Roman of his time. The question he asked could I think be asked as well of Christianity, which through Paul and his followers came to have less and less to do with Jerusalem. I'd say Christianity as we know it has been more influenced by pagan thought and religion than it has been by Judaism, with which it has always had a rather awkward and sometimes violent relationship.
So I think I give the Church its due, in that I acknowledge that it kept a great deal of the pagan West alive through its assimilation of it. And I don't mean to contend that the "barbarians" as they were called by ancient Greeks and Romans didn't contribute to Western civilization. But I think Western civilization to the extent it can be said to be of the European tradition looks back to the Roman Empire and Republic.