The Pope's dominance only started because he was left alone in the West as Imperial power withered and Islam conquered three of the other Pentarchs.
Yes that is actually a very interesting & intriguing subject. Initially (after the fall of Western Rome that is) the Bishop of Rome & the western Nicaean Church was guaranteed protection by The Eastern Roman/Byzantine Empire. As the ERE was still the undisputed superpower in the Western World in the 5th/6th century and still operated under Emperor Constantine's old "One Emperor, One God, One Empire" philosophy - for instance the Barbarian kings in the West would not dare assume the title of Emperor and were given the title of Patrician or Consul if they submitted to the authority in Constantinople.
However there is a few contributing events that seemed to split Rome from it's more Orthodox roots.
1) Obviously the decline of classical civilization: At its peak trade and the sharing of ideas and culture kept the Roman Empire 'Roman' and this was true for the church. The first Ecumenical Council was called to solve disputes - at this time there was really only 1 church and bishops could be called from everywhere to attend a single meeting.
After the fall of Rome, East (Greek speaking) and Rome (Latin Speaking) became separated and due to growing language, cultural and distances the Roman church was isolated from the Greek speaking East which really was the heartland of Christianity in that period.
2) During the barbarian invasions Augustine of Hippo wrote many theological doctrines which strongly influenced the teachings of the Western Church. As the Western church was cut off from the East at this time (due to the barbs) these ideas were never debated in an ecumenical council. Hence some doctrines in the Catholic church particularly Original Sin can be traced to Augustine.
This is important because exploring christian concepts like sin, human nature, and the state of the soul goes to the heart of the christian faith. E.g. is the body inherently wicked - as Calvin later taught - (he in turn was heavily influenced by Augustine's writings) or is human nature good & made in the image of God but now under the influence & power of sin because of the fall...... (from what I gather the latter is sort of how the Orthodox Christians view it which is very different to that of Roman Catholicism but I'm not good at trying to explain the details.
But to summarize I think it was the inclusion of many of Augustine's teachings and he was heavily influenced in his early life by pagan philosophies. He also never completely mastered Greek, and that was a problem because much of Christian philosophy had been written in Greek.
The original biblical manuscripts were also in Greek so without a full mastery of Greek trying to copy a Greek manuscript into Latin without introducing error is probably impossible and it is likely that he introduced some ideas into the Roman Church that would have been rejected in a full ecumenical council.
3) The rise of Islam is probably the most important factor in what led the Roman Clergy to swing toward a military state. The spread of Islam co-incided with the decline in power and influence of the Eastern Roman Empire and probably was the main reason why the Bishop of Rome crowned Charlemagne as Emperor. So now the Pope had authority over political statesmen.
Saint Peters Basilica in Rome (which was outside the walls) was actually raided by Arab muslims in 846.
Around this time Byzantium was probably at it's weakest point and the Franks were growing in strength so the power dynamics in Europe were changing enough that the ERE wasn't the only power that the Bishop of Rome could rely on for support (especially at that time because Justinian's 6th century re-conquests into Western Europe were nearly all lost).
It was actually the Pope that took authority of Rome during that crisis in 846 in which case the church can hardly be blamed as there was no secular power in the region that could protect the church at that time - so the Pope had to assume authority because no-one else was able to help. So the Vatican's move to being a military power started as a reaction to the aggressive spread of Islam through the Mediterranean and the importance of this can't be understated. For centuries the Mediterranean Sea was the lake of the Roman Empire and was central to commerce. Once the Islamic Caliphate conquered the ports in Syria and Egypt muslim pirates went out and plundered, raped and enslaved the people in virtually every bit of coastline they could find. This had a massive impact on society and signaled the end of late-classical urban culture as people sought refuge behind walls and castles for safety. Trade also disappeared as the silk road was cut-off and people were largely reduced to subsistence living.
4) Iconoclasm would also have played a part and this period which was later regarded as a heresy by the Orthodox but it would have strained relations at the time. Rome actually excommunicated the Iconoclasts.