Quoting from the FAQ: "Europe was in the right place, at the right time, with the right circumstantial history, and the right systems in place that when the Industrial Revolution happened, its hegemony was cemented."
I'd like to know one of the aforementioned right systems and right circumstantial history could be traced back to the Middle Ages and even the so called "Dark Ages". I would be also curious on the importance of the Catholic church in this process. I think that the cohesion of one religion and the competition among many nations gave Europe a good defense against external threats without stalling its development, alas I have not sufficient knowledge to back this idea or explain it further.
I think I know what you're grasping at, so let me suggest that before you think of what a "European hegemony" means at the same time you bring up the Catholic Church, you first read up on the Investiture Controversy. There is a fantastic read here that focuses on the reform aspects.
One key question there was, who was the ultimate representative of God's power on earth? Was it the pope and his church (sacerdotium)), or the emperor of the Holy Roman Empire and his secular rulers (regnum)?
The Holy Roman Empire itself is a magnificently strange beast. Prior to 800, it was simply called "The Empire". There was only one Empire in Christendom, so why bother with anything other than "The Empire"? Clearly, there was much hewing to a hegemony -- a Christian hegemony -- even if it wasn't clear how exactly it was supposed to work. Was it a replica of the Roman Empire? Was it a new empire?
In 800, Charlemagne had been granted the title Imperator Romanorum By Pope Leo III, but he didn't use it lest he offend the real Roman Empire in Byzantium. His later successor Otto III started to use the title consistently, and in 998 adopted the motto Renovatio imperii Romanorum. A convenient political statement perhaps, but he was not much closer in restoring the Roman empire than anybody prior. Over time, however, successive emperors started to use the title more consistently and over time it became the norm.
At the same time, the Popes in Rome happily played along. They wanted a useful-but-distant emperor, lest the emperor meddle in the actual business of ruling Rome.
So between the emperors and Popes came a complex charade wherein the emperors use the prestige of the Holy Roman Empire and papal blessing to strengthen their rule, and the popes keep these emperors at arms' length. Or in most cases, arms long enough to keep them mostly north of the Alps.
The investiture controversy resulted in wars (surprise!) and the outcome of that conflict was a compromise that strengthened church authority and weakened the emperor. Emperor Henry V allowed for free election of bishops and imperial abbots, and Pope Calixtus II allowed the election of prelates in the HRE to be in the presence of the king (or emperor).
The result of this is that the idea of a universal church prevailed but the idea of a universal empire was weakened.