Rome was last United under Theodosius The great (the ONLY Roman Emperor ever called "Great"). He re-united the Empire by defeating the Goths after the Roman Military Defeat at the Battle of Adrianople.
Theodosius I:
http://en.wikipedia.org/wiki/Theodosius_IBattle of Adrianople"
http://en.wikipedia.org/wiki/Battle_of_Adrianople_(378)
Now, his method of defeating the Goths were unique for the Roman Army, he did NOT defeat them in any one pitch battle, instead defeated them piecemeal as he rebuilt the Roman Army after it was almost Completely wiped out at Adrianople. He then defeated the remaining Goths and incorporated them both into his Empire (Into what used to be Yugoslavia). He then completed the transformation of the Roman Empire into a Catholic Empire (Through paganism was NOT outlawed, it was tolerated and would be for another 200 years, except for the occasional sacking of Pagan Temples when Gold Coin became short, the Gold in the temples went the Emperor, the Temple to the Christians).
At the death of Theodosius I in 395, he divided his Empire between his two sons, and told them to work together, which they did. The problem was the Empire was heading in at least three different direction at that time. The West, Egypt and Syria (Egypt and Syria spoke a variation of modern Arabic) was going further into the rich getting richer and the poor getting poorer. The East (i.e. Greek Speaking part of the Empire) was the last stronghold of small farmers. Theodosius continued Constantine plan of using Christianity and its network of churches to unite the Empire, both Rich and Poor, while minimizing the conflict between Rich and Poor. This worked to a degree, but in the West after about 420 (i.e. the time of Attila the Hun) the Rich ran out of money and turned to the Germans as troops to keep their own peasants in line. The problem was the Germans were peasants themselves and after a few years embraced the peasants against the Roman Elite and started land reforms (Which Romans had needed since the time of the Gracchi).
More on the Gracchi:
http://en.wikipedia.org/wiki/GracchiAnyway, as the Germans imposed land reform, the Romans Re-invaded under Justinian (who had also outlawed the last Pagans holdouts in the Empire). When the Germans Imposed their Land Reforms (Opposed by the Roman Elite) the Church is conspicuous by its absence in the official record BUT then is seen as allied with the Germans and Peasants, through also allied with the Eastern Empire under Justinian. The best explanation is the Church provided the background for the land reforms (i.e. did the actual division of the land) while the German occupiers ordered the Land Reforms while the Roman Elite opposed it (All Complaining to the Eastern Emperor after 450 AD).
Now, the Church had assumed most of the domestic functions of the Roman State as the Western Empire broke up. The Break up of the Roman Empire is a better way to view what is traditionally called the Fall of the Roman Empire. Except for the Vandals, ALL the Germanic tribes that invaded the Empire were defeated by the Roman State, but then permitted to live in areas where they had been extensive peasants revolts against the Roman Elite (One of my favorite is the Revolt of Armorica, Modern Brittany, where the Roman Sent troops, after the death of Attila, When Roman power was in an upswing, rather then the Goths and Franks who were closer, for the Armoricans were a peasant revolt that had been successful in taking land from the Roman Elite, while the Goths, Franks and other Germanic tribes had not yet done so and would NOT till the late 400s and the 500s). While Roman Military power was on the decline, most People still called themselves Romans. As Romans they were subject to Roman laws, while the Germanic Tribes were subject to Germanic Laws. This created a dual legal system that lasted till the Middle Ages, one for "Romans" and one for whatever Germanic Tribe one was a member of. The Peasants had been Romans and while they like the land Reform being done by the Germans, they viewed themselves as still being Romans. As the Military Power of Rome withdrew to Modern Greece and Turkey (The EAstern Empire would lose Syria and Egypt in the Mid 600s) people in the occupied areas still called themselves Romans. The only institution that survived the break up of the Roman Empire was the Catholic Church, thus the Romans who no long lived under the Roman Empire only had the Church to help them defined themselves as Romans. Keeping track of Births and Deaths, marriages and personal property, which had been done by the State as long as the Roman Empire Survived, had to be performed by someone and only the Catholic Church had the knowledge of Roman law, and writing to do so. Thus people embraced the Catholic Church as their Church and how they identified themselves as opposed to the Germanic Tribes who technically controlled all of the land and provided the military force for the area.
AS time went on, even the Germanic Tribes embraced the Catholic Church for their wanted to be one with their people, and their people were Romans and Catholic, so the Germanic Invaders slowly became Catholic and "Roman". This was NOT a quick change, even as late as the Renaissance certain groups claimed non-Roman Citizenship based on old Germanic tribal rights, but by that time did not last long.
Thus by the "dark ages" most of Europe still considered itself "Roman" (The Main exceptions were the Germans in present day Germany and the Slaves in Eastern Europe, but both groups came under pressure to join the rest of Educated Europe and become "Roman" in outlook as their embraced Catholicism). Furthermore Every Bishops had to make a trip to Rome to talk to the Pope (The big exceptions were the Bishops under the Orthodox Patriarchs, which included modern Greece, Turkey, Syria and Egypt as after its Conversion Russia). Thus the best way to get messages to each other was through the Catholic hierarchy. Furthermore this is the days before linen paper, so paper was expensive (Parchment was the paper used and it is animal skin). Thus if you wanted to get messages to everyone in your kingdom you had to send it through the bishops and the priest for there was no other way to get the messages to the people. Thus the Roman Church Survived based on its control of communications. As long as no one could challenge the Church's control of Communications it had power, and in most people's lives more power then the local king for all he controlled was his troops (and most of them could only be called up via the Church and its communication system).
The introduction of linen paper in the 1300s into Europe made paper cheaper, this actually enhanced the church by permitting more sending of letters then having to do many messages verbally. On the other hand, groups outside the Church could also afford the new linen paper and start to send messages to each other, but it was still quicker and cheaper to send messages via the Church then any other way till the mid 1800s when Pulp paper came into widespread use and with it what we would call "Newspapers" (Newspapers existed before that date, but their were more advertising rags then what we would call "Newspapers", Modern Newspapers did not spring up in the 1800s, many trace their roots to the 1700s, but till the 1800s, pulp paper, steam locomotives and high speed presses, modern newspapers were NOT possible). I go into this for the survival of the Roman Catholic Church is more a product of the need to get information to people then anything else. The Catholic Church, based on its hierarchy, which had been adopted in the 100s as a duplication of the Roman Imperial system, but extending it down to the peasants on the streets, was the best way to get information to people and thus was the best means of Communications. The Catholic Church maintained its Roman structure and even expanded it into Germany and the Eastern Slavic nations and as long as it was the SOLE means of Communication it survived and with it its Roman Traditions and background (yes it survived today, but with Modern Newspapers, Radio, Television, and now the Net not as strong as it was just 200 years ago).
Sorry about the rambling, it is pass my bedtime, but I felt the need to file some-sort of answer. The rambling is more the result of the reason the Catholic Church is the heir of the Roman State is because the Catholic Church (And I have to include the Orthodox when I use the term Catholic here) is the only institution that survived the breakup of the Roman Empire between 450 and 700 AD. Those 250 years, from the time the real ruler in the Western Empire was NOT the Emperor but his German head of his troops to the defeat of the Arab attack on Constantinople, saw the Empire go from a country speaking three different languages (Latin, Greek and Arabic, through the Romans would have called the last language Egyptian) to one that spoke only Greek. The empire went from an Empire of large estate owners, owning vast estates from Britain to Egypt, AND a mercenary army to a state made up of small farmers who in exchange for their land had to provide military service to the state (Yes the Byzantine Empire, after 600 AD, did NOT have the same number of wealthy elite that the Roman Empire and its successor states had till about 500 AD, no one knows where their went, most of the old Roman Families tend to just disappear through some held on till the 900s).