The history of the Roman Catholic Papacy of course is long and convoluted, but we can briefly touch on various events, time markers, and players on the stage during the first 500 years of Christianity that help shape and define the idea.
Let’s start with a definition of Papacy. The Papacy is the office of Pope of the Roman Catholic Church extending from the apostle Peter to the present Pope. As such, the Pope is the supreme head of the whole Church with complete authority over matters of doctrine and practice (faith and morals). The Papacy relies on the ideas of Peter as “rock”, a unique power to “bind and loose”, and possession of the “keys to the kingdom”.
Was there a time when most of Christianity recognized this supreme authority? Kind of, but not really. Sources follow comments.
As mentioned, the Papacy took time to fully develop, but hints of it surfaced within the first generation. Clement of Rome wrote an epistle circa 95 CE that suggests that one "tribe" will have a priestly authority, as Levi did over the other 11 tribes. The power to forgive sins, of course, is the whole point of authority.
In 155 CE, Anicetus of Rome and Polycarp of Smyrna argued over the observance of Passover (aka Easter). In 175 CE, Victor of Rome attempted to excommunicate all of Asia Minor led by Polycrates over the same issue. It was not until the Council of Nicea ruled with Constantine as its guarantee in 325 CE that the Roman view of Easter prevailed. It took the State’s authority to force a certain spiritual authority at this early stage.
Tertullian circa 195 CE follows in the footsteps of those who disagreed with Rome’s budding authority. He first mentions how Rome was usurping the authority given to Peter alone. Cyprian circa 220 CE will quickly follow, but in a positive note of Rome’s authority; that is, it is necessary as a visible unity of the Church. In the next breath, however, Cyprian will maintain the right of the bishops to disagree with heresy wherever it arises, as they did over the validity of heretical baptism.
Evidently it was in Leo the Great, however, when the full Papacy idea had completely blossomed. This was circa 450 CE. This is the time when even Alexandria will fully comply and acknowledge the Easter times of Rome. The argument used was how could Mark, Peter’s successor, teach something in Alexandria that was not taught in Rome? Disagree and there goes one’s claim to being an apostolic church.
Besides, Pope Leo successfully persuaded Attila the Hun from invading. Who could argue God wasn’t on his side?
The doctrine or more loosely the idea of “papal supremacy” residing in Rome predates Nicea. We clearly see Pope Victor of Rome exercising this supremacy, although the rest of the Church disagreed with him. We also understand Tertullian and Cyprian seeing Rome claiming this authority. Again though, while Rome may claim it, the rest of the Church does not completely agree with it. One may claim supreme authority, but that doesn’t mean one checks their brain at the door against heresy.
Yet in Leo the Great, the Papacy is fully known and exercised with at least all of the Church submitting. No doubt individuals would disagree, but the patriarchs of Alexandria, Jerusalem, Constantinople, and Antioch would evidently all submit. Even still there was one exception to this authority who came to be the Oriental Orthodoxy. This was “sub groups” within a couple of the patriarchs. Of course, 500 years later, the Eastern Orthodox will also disagree. 500 years after that, the Protestants also disagree.
Rome clearly believes in its Papacy. It has done so for some 2,000 years as the idea of an “authority” over the whole with binding and loosing powers and the keys to the kingdom surfaces within the first generations after the final apostle’s death and carries through today. The rest of Christendom, however, never really agreed with the view from Rome.