As the Muslims took over everything south of Europe, the Roman Catholic church sent missionaries to Ireland, Britain, Germany, Spain, other parts of Italy and other Nordic countries. Patrick went to Ireland, Columbanus went to Switzerland, Columba to Scotland.
Boniface, aka, Winfrid, evangelized the Teutons in Germany, Wilfrid, went to Friesland (Netherlands), Gregory I converted the Lombards away from Arian "Christianity," and Christ came to Spain although the Spanish also suffered confusion from the Muslims.
What I find interesting about the Catholicism that spread to Ireland, is that it was not necessarily Roman. Monks married, they did not always follow the pope, they had different haircuts, and these confusions bothered the Islands. Germany converted when its leaders accepted Christianity. Most of them based their belief on their leaders. The Lombards came to the real faith, and the Spanish were taunted by Islam. All of these eventually accepted unity under the Roman bishop.
People will deny it, but Christianity has always been divided into denominations. Some have serious differences, some are just matters of interpretation. Paul noticed them in writing to the Corinthians. The world was divided among 5 major bishops and the Roman bishop grew in power while the Eastern bishops sunk into heresy. It seems also the countries touched by Rome had differences among each other that eventually became cemented when they took the leadership of the Pope who claimed that his Church alone has the keys to the kingdom.
To an extent, men like Gregory the Great were right. There is no salvation outside of Christ's Church. You need to belong to a visible church body and all the church bodies must agree on all the important doctrines. These bodies are ruled by men who interpret Scripture and teach it to the people. At this time, however, not many people could read the Scriptures, and they had to rely on the leaders for their spiritual nourishment. The leaders, being human, sometimes interpreted Scripture wrong.
Jesus did give Peter the keys to the kingdom of heaven in Matthew 16. He said on this Rock he'll build his church. The problem comes in when people misinterpret said Rock. Jesus gave Peter access to heaven because he openly professed Jesus to be the Christ, the Son of the Living God. Jesus is the Rock on which he builds his Church. He is the leader, the chief priest, and our ultimate authority for life which people can read in the Bible. Somehow, people interpreted this passage to mean that Jesus based his Church not on what Peter said, but on Peter. Then, Rome claimed to have its bishops in succession from Peter.
However, the Church was not built to follow one man's interpretation. Jesus appointed many apostles who are succeeded by preachers today who are supposed to keep each other accountable to what Jesus teaches in his Word. Somehow, this accountability became concentrated on Rome until they believed that only their theology was the right one, even when it began to contradict Scripture and impose rules never intended by God, such as priests having to be celibate.
Then again, this is just a repeating picture of what always happens in Christendom. Rivals show up both outside and inside the church, and people do not verify their facts with Scripture, or as in this time, an accountability system of many men who preached Scripture. And this is when the Church focuses its allegiance to one man, his ideas, and follows this man without questioning whether he agrees with Christ or not. And this happens in all divisions of the Church. We have to remember that Christ is the head of the Church. He is Adam, the Church is Eve, and he leads us. While the wife is on the earth keeping house, she should never forget her husband and that he leads the family.