I am well aware that during this period there were many scientific and technological achievements in many parts of the world, including the Islamic World, the Late Roman/Byzantium Empire, and the East Asian cultural region. However, in terms of Western Europe’s “Dark Age,” exactly how responsible was Christianity in causing this? Would the “Dark Ages” still have happened without the advent of Christianity?
They did indeed have a role in the "Dark Ages", primarily in staving it off. The term "Dark Age" refers to our (at the time) lack of historical information as to what was going on. One of the major roles of the Church at the time was the storing of historical information.
Thomas Cahill, in his 1995 book How the Irish Saved Civilization:
[A]s the Roman Empire fell, as all through Europe matted, unwashed barbarians descended on the Roman cities, looting artifacts and burning books, the Irish, who were just learning to read and write, took up the great labor of copying all western literature - everything they could lay their hands on. These scribes then served as conduits through which the Greco-Roman and Judeo-Christian cultures were transmitted to the tribes of Europe, newly settled amid the rubble and ruined vineyards of the civilization they had overwhelmed. Without this Service of the Scribes, everything that happened subsequently would be unthinkable. Without the Mission of the Irish Monks, who single-handedly re-founded European civilization throughout the continent in the bays and valleys of their exile, the world that came after them would have been an entirely different one-a world without books. And our own world would never have come to be.
Church records of births, deaths, and marriages continue to be used to do historical analysis of the prevalence of death and disease during that time period. The preservation of literature by the Catholic scribes has provided many of the extant documents of the time. For example, Boethius's Latin translation of Aristotle's treatises on logic were the only version available through the 12th century according to Robert A. Guisepi's Europe's Search For Stability. They maintained the schools and libraries ("Woods, How the Church Built Western Civilization (2005), p. 40" and "Le Goff, Medieval Civilization (1964), pp. 80–2") of the time and founded the first universities.
While I recognize that it's not likely how you intended the use of the word, I'd argue that historical evidence is that the Church were some of the most responsible people involved in the ending of the "Dark Ages" by ensuring that knowledge was not lost.
Maybe you are forgetting that, if Christianity had never existed, Europe would not now look like it is today. After the fall of the Roman Empire, people were constantly attacked and plundered by invaders coming from the East and the North. If monasteries had never existed, two consequences would have likely occurred:
- most of the progress and culture gained during the Roman Eve would have been completely lost
- Europe would have become a simple colonial territory to be exploited by all people passing by
Men like St. Benedict and St. Patrick are not only famous because they were monks. Instead, first off they wanted to be happy themselves. What they did was just to look for where they could find their happiness. As they found it, then they could no longer keep it for themselves only, but they had to tell other people that there was a better way to live. This is why monasteries were founded and towns and entire nations (!), which is the case of England and Ireland, grew upon the monasteries. Monasteries quickly became the main cultural and political centres not because monks wanted the power for themselves. Instead, people realised that they were the only ones who could help them carry on during those hard times, where wars and famines were routine. Don't forget that if monasteries had never existed today we could never read 95% of the Roman literature, as well as very few people would be able to read. Ah, just another thing. If Christianity had never existed, structures like hospitals and hotels like the ones we conceive of today would not exist.