"Papal" State implies no or very late reunification with East
When Christianity was adopted by Emperor Constantine, formalized at the Council of Nicea as far as the newly-organized Church then was concerned, he took care to make himself, as Emperor, the supreme power on Earth of the Church hierarchy. The various Patriarchs then established were clearly meant, as far as the imposed formal administration was concerned, to be equals and, as far as one person could be said to be over another, subordinate to the Emperor.
There were even then nuances of course; the Church deemed itself no new creation of the Empire but the order established by God Himself on Earth, extant in worldly terms from the time of Christ and in a sense for all eternity, and ultimately far greater than any worldly thing like an Empire. Of course Christendom, in the sense of people who held themselves to be Christians, was not even in skeptical and worldly historical terms a creation of the Emperor; Christian peoples existed already both as, until yesterday, illegal dissidents within the hitherto pagan Empire and as residents of societies outside the Empire, some of whom would later invade and bring their own doctrines deemed heretical at Nicea, others of whom never joined the Empire in any sense. Of course Nicea did not settle the matters of true faith and doctrine to the full satisfaction of even all the faithful Christians of the Empire, let alone the world.
Insofar as any particular authority on Earth other than the say-so of Emperors and later, Patriarchs (of whom the "Pope" was merely one, the Patriarch of Rome) existed, it would be Councils such as Nicea, gatherings of large numbers of Church officials to meet together and work out the true doctrines that in principle all Christians were supposed to adhere to. This principle actually prevailed even in the Roman, Western sphere; it was only at Vatican I in the late 19th Century that the Roman Catholic Church determined that the Pope was supposed to be infallible in matters of faith and doctrine and then only under specifically defined conditions ("speaking ex cathedra"). I believe that in principle a Church Council is actually implicitly still held to be the real final authority, short of God Himself speaking clearly and directly, by Roman orthodoxy.
But in reality of course in the early Christian years of the Empire it was the Emperor who in fact held the last word--insofar as he could enforce it! Dissent between the declarations from Constantinople and the locally-held dominant views of the distant diverse Patriarchates of the Empire, even at the height and breadth of Imperial power, were tremendous factors in Imperial history and for instance account for much of the success of the early Muslims in overwhelming and removing huge parts of the Empire from the Emperor's authority.
OK, to bottom-line all this historical background--the Patriarch of Rome was never, in the years of the Empire's strength, even considered first among the Patriarchs; the controversies tended to be among the eastern regions, with Rome's sphere quickly slipping from Imperial control except in small remnants and rarely and briefly coming back under it; the Roman sphere did well to survive at all and no one in the richer, more populous, more central in every sense East looked to Rome for advice. It was only over centuries that once were simply called "the Dark Ages" in our historiography that the Patriarch of Rome came to be considered the Patriarch, the last link of the chain up toward God and Heaven who lived on Earth, and calling "The Holy Father," affectionately diminished to "Il Papa," "The Pope" in English, arose only among the long-benighted and politically, culturally, economically chaotic Westerners who heard nothing at all from the East that had written them off long ago.
So--for, at some later date, "The Pope" to be deferred to as the supreme voice of God on Earth, can only refer to the "Latin" westerners. If as in some replies to this thread, the reunification of Nicean Christendom were accomplished early in the second milennium, I don't think the Pope could retain his unique dignity, at least not among the reabsorbed Easterners who had never lived in Rome's defined sphere. Even if the reunification of some great Empire were accomplished mainly by the high-handed efforts of Latin Westerners who somehow (this part is very unlikely!) had the ability to run roughshod over something like a thousand years of divergent local custom and impose a Roman rite on the whole sprawling domain, I'd think that eventually, within a few generations at the most, local powers there, even ones of thoroughly Western origins, would find it politically expedient to revive and capitalize on ancient resentments of Roman pretension and promote the equal authority of local patriarchs, resulting either in a new schism where the Empire breaks up again into rival Christian powers, or the re-invention of Constantine's scheme whereby the supreme secular power, presumably calling himself "Emperor," again takes his place as overriding any and all Patriarchs for the good of Christendom and humanity as a whole. The West, even if it remained broadly loyal to Rome's brand and grew as OTL relative to the East in power and population, would not outvote the East, however abject the original relationship in the restoration might have been.
So--"Papal" State implies that in fact, the Roman sphere either never does re-absorb the East, or does so very late, long after the old continuity with the Constantinian hierarchy had been well and truly broken down by subsequent failures of the Empire there. (If that doesn't happen, and yet a Papal West does manage to subordinate an East that kept that continuity, we eventually have the shift to the East I outlined above, or the system breaks down completely before that can happen and again we have the Roman Church in the west in schism with the East).
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Can such a Papal State, based on Rome, have existed? Well, I suspect that having it rise to be basically tantamount to an Italian realm seems unlikely. OTL, the small Papal States arose mainly because as a practical matter, the Church in Rome had to secure its own house politically just to maintain a dignified independence of whoever else happened to locally run that part of Italy. The whole idea of elevating Charlemagne to the dignity of "Emperor" was to hope to create a power that would reliably guard the Pope and free him up to focus on the spiritual guidance of all Christendom (as the Roman Church deemed it), but in fact of course it wasn't long before the ideal balanced relationship between Pope and Emperor broke down and soon Popes were seeking allies against the current Emperor. For a line of Popes to successfully secure a solid base in Italy and then build that up into a solid realm run as their own secular state would throw the dignity of the Roman pretension to stand at the head of all Western Christendom as the universal shepherd into the vortex of princely power politics. In fact of course it was right there anyway in OTL and hence the spectacle of Anti-Popes and purely political schisms that were eventually resolved at sword point and negotiating tables. But I suspect the limited, puny size of the Pope's direct realm was no accident; it was the only empirical solution--give the Pope enough to buffer himself from the petty schemes of local politics but not enough to set himself up as a player of grand secular games. So, Papal Italy seems an unlikely outcome--if a line of Popes determined to achieve this succeeded, they'd also pretty much guarantee that there would be Anti-Popes set up outside in rival nations, and these schisms would persist as long as Papal Italy did, splitting the formerly "Roman" world into a new, shattered matrix of national patriarchies. Soon, the only distinction between Papal Italy and other kingdoms would be that they claimed to be a theocracy rather a secular dynasty--but that would hardly be unique; medieval and Early Modern Europe abounded with ruling Archbishops running substantial realms from a city-state base, presumably even Bishops that claimed communion with the Roman power would use their scattered isolation as diplomatic leverage to jump themselves up to de-facto status as yet more Church-run states.
So that's one scenario--"Roman Catholicism" becomes more honored in breach than observance and the Pope is just another theocratic prince among many.
Another would be, that the Popes learn to specialize, as OTL, in negotiation and balance among the many secular powers of Western Christendom, and achieve their political bastion by playing them off against each other skillfully (more or less!) rather than by trying for some large and growing territorial basis they run outright and directly. One way I could see this diverging from OTL would be if the Holy Roman Empire and its more or less successors, the Hapsburgs, fail early and spectacularly to keep the mantle of "protectors of Christendom in general" and by default, that role falls to the Popes, as overseers and organizers, who then manage to play it successfully. It could be that whenever the region as a whole comes under strong external threat, the Popes manage to organize a collective response other than creating a specific new secular state to do the job. Such as, calling a Crusade and negotiating truces between neighbors hitherto at each other's throats, and collecting detachments of force from all powers in a league they manage, to counter the threat and perhaps push it back. Then, rather than use this organization to simply create a new Empire (which would either eventually break down as OTL, or if successful would soon overshadow the Papacy) they take charge of maintaining its command nucleus while carefully keeping the nominal and to some extent practical independence of the various kingdoms and city-states and leagues and so forth in place, to check one another. In short, something like a Catholic United Realms, a sort of churchly analog to the US federal government, might be the "Papal State."
Such a federal union might eventually evolve into a strong central Empire, one whose sovereign would be the Pope rather than a secular dynasty or elective Emperor.
But to conclude, I don't think we could have this happen on a smaller scale than all Roman Catholic Europe, because any nations not included would eventually cease to be in communion with Rome. And I don't think it would stay Roman Catholic if in a fit of enthusiastic energy, a bunch of Latins overwhelmed the East early on--and still less if reunification with the East happened more consensually.