Someone suggested the Roman Catholic Church began with the Edict of Thessalonica in 380 cherry-picked a quote to prove it. Here is a quick but much more accurate description of the origin of the Roman Catholic Church.
The Theodosian Code is a big collection of edicts that was made by Theodosius II in the 5th century. The specific quote you gave is from the Edict of Thessalonica in 380. Here’s the full text (from wikipedia:
It is our desire that all the various nations which are subject to our Clemency and Moderation, should continue to profess that religion which was delivered to the Romans by the divine Apostle Peter as it has been preserved by faithful tradition, and which is now professed by the Pontiff Damasus and by Peter, Bishop of Alexandria, a man of apostolic holiness. According to the apostolic teaching and the doctrine of the Gospel, let us believe in the one deity of the Father, the Son and the Holy Spirit, in equal majesty and in a holy Trinity. We authorize the followers of this law to assume the title of Catholic Christians; but as for the others, since, in our judgment they are foolish madmen, we decree that they shall be branded with the ignominious name of heretics, and shall not presume to give to their conventicles the name of churches. They will suffer in the first place the chastisement of the divine condemnation and in the second the punishment of our authority which in accordance with the will of Heaven we shall decide to inflict.
In the late fourth century, when this was written, the honor of the bishop of Rome was at a peak. Constantius II (Constantine’s son) had ravaged the eastern churches, replacing Nicene bishops with Arian ones as often as he could. Rome was the only apostolically founded church that was out of his reach because his brother Constans reigned in the west. As a result, Julius of Rome was the great champion of Eastern bishops that had been ousted by Constantius. That mess continued until Emperor Theodosius I cleaned it up between this edict and 383.
Also, by that time, it was believed by all bishops that the unity of the Church was based on Peter. Cyprian, in the 250s, had taught that there was just one episcopate (one office of bishop) shared by all bishops. This unity of the bishops was the unity of the churches, and they all had inherited this from Peter. Late in the 250s, a bishop name Stephen arose, who counted himself the sole heir of Peter’s authority. He excommunicated churches who disagreed with him, but every one opposed this action even if they agreed with him.
The point of telling you about Cyprian and Stephen is to give the origin of the idea that the bishop of Rome was the sole heir of Peter’s role as representing the unity of the Church. Really, Cyprian is the earliest known write to say that Peter was the foundation of the unity of the Church, but he was talking about all bishops, not just Rome. Then, 70 years later came the Arian Controversy and the Council of Nicea. The council did not solve the controversy only because Constantius II became emperor in the East in 337 and took up the Arian cause. He unfortunately reigned for 24 years and really caused problems in the Eastern churches. He actually became sole emperor in 350, I think, and for a decade he caused problems in the West too. He even got Bishop Damasus of Rome to recant the Nicene Creed by imprisoning him.
Anyway, by the time this decree came out, it is no surprise that an emperor (or three emperors, in this case) would write that the faith was delivered to the Romans by Peter. It does not, however, mention only the bishop of Rome, but also Peter, bishop of Alexandria. It then gives a quick description of the Trinity in line with the Council of Nicea. Under Theodosius I, the Arian Controversy would be mostly put to rest. It was only a minor annoyance after that.
This is in no way the origin of the Roman Catholic Church. That is a development. For there to be a “Roman” Catholic Church there had to be a split between Rome and the rest of the “Catholic Church.” Nowadays, the rest of the Catholic Church is usually understood to be the Eastern Orthodox Churches. The split between them and Rome happened officially in 1056. That could be said to be the origin of the Roman Catholic Church as opposed to the the rest of the Catholic Churches.
We must remember that the Roman Catholic and Eastern Orthodox Churches only represent the Roman Empire (and later, Russia). The churches in India have their own “Catholic Church,” as did the Persian Churches, now part of the Oriental Orthodox Churches. It’s all complicated and the divisions between the various Orthodox/Catholic Churches happened at various times, and some have reunited in the last few centuries.
Anyway, the origin of the Roman Catholic Church was not some moment.. It was a process of separation from all churches outside Europe and a growth of power in Rome that began in the fourth century and did not culminate until the eleventh.