"Civilized" is a problematic term which historians don't really use.

By fall of rome I assume you mean the western Roman Empire. The eastern Roman Empire, called the Byzantine empire by modern convention, continued for many centuries after the fall of the west. Even before the west fell, the east had surpassed it in terms of wealth, power, and cultural infuence.

As /u/blackflag415 noted, this question is pretty hard to answer. Aside from the difficulty associated with using the term 'civilised', 'prosperity' is also quite hard to pin down given the lack of solid economic data. It is perhaps possible to answer this somehow based on which state was perceived as the most 'civilised', but the attitudes of the sources towards each polity can and did shift over time, so our subjective measurement of this can vary quite dramatically. My comment here will mostly be a dissection of this question, as I don't think there can be a proper answer to this question, though I am happy to concede that there are other viewpoints on this. I'll also largely ignore the economic side of things I'm afraid, since I'm not particularly well-read in that area.

The answer regardless will probably be the Eastern Roman Empire though, since due to its size and possession of wealthy provinces like Egypt it was easily the richest polity in the Mediterranean world, and as contemporaries judged how 'civilised' something was through a comparison to how Roman it was, the Roman Empire was by definition the most civilised thing around. Persia probably didn't see it this way, but by this point Rome and Persia had developed a form of ideological detente - both saw each other as inferior, but in practical terms both empires were happy to co-operate on most matters and a very elaborate diplomatic system had developed by the sixth century. Eventually the ideological accommodation between the two culminated in this wonderful line from the early seventh century:

God effected that the whole world should be illumined from the very beginning by two eyes, namely by the most powerful kingdom of the Romans and by the most prudent scepter of the Persian State. For by these greatest powers the disobedient and bellicose tribes are winnowed and man’s course is continually regulated and guided.

For all the traditional bluster and propaganda from the Roman side, the Persian 'barbarians' weren't too bad when compared to the 'real barbarians'. This is especially notable because this was written after 26 years of war between the two empires, so it is quite clear that the Roman state was trying to spread the idea that the Persians were tolerable after all. The same was probably the case from the Persian side, but unfortunately we don't have any sources from the Persians to confirm this. This also brings up to the more general point I want to make here, which is that being 'civilised' or not was a hugely flexible concept. Romanness was actually pretty negotiable - emperors can be called reges/kings, whilst a 'barbarian' king can also be proclaimed as an imperator or an augustus. Many proper emperors, such Zeno and Justinian, were hardly shining examples of Roman culture. The former was an Isaurian from one of the least Romanised parts of Asia Minor, whilst the latter was born a peasant - how were they any more Roman than the Germanic 'barbarians' in the west or the Arab frontiersmen guarding the eastern frontier? Late antiquity was a period of dynamic cultural interactions and putting people into different boxes isn't the way I'd approach this topic. Romanness was widely lauded, but it also changed constantly and was used differently by different people - who are we to judge what was 'properly' Roman or not?

If I have to making a sweeping judgement, I'd probably say that almost everyone within the boundary of the empire saw themselves as Roman or successors to the Romans. The former province of Britannia was the exception, as we literally have no written evidence of what people were thinking about themselves at this point; we have one source, an angry sermon from Gildas, but it doesn't tell us all that much, though luckily given the province's remoteness it was never in the contention for a particularly prosperous place anyway, even if we ignore the internecine warfare and economic decline that characterised the province in late antiquity. Elsewhere though it was rather complicated, since no-one liked to think of themselves as 'barbarians' and new polities emerging from the ashes of the empire all utilised Roman institutions as much as possible - taxation, law-making and Roman culture were all useful, so no-one felt a particularly need to get rid of them. Perhaps more importantly, all 'barbarian' groups were outnumbered by the local Romans they ruled, so a slash-and-burn approach to government would have failed spectacularly. Instead, western kingdoms all acknowledged the power of the emperor in Constantinople and did their best to secure his favour. Fancy titles, such as the consulship and magister militum, were regularly requested (and granted) and coins would continue to be issued in the emperor's name; the first exception was Theodebert I of Merovingian Gaul (533-548), but even then he just took the imperial template and replaced Emperor Justinian's name with his own. On his coins he described himself as 'Dominus Noster Theodebertus Victor' (Our Lord Theodebert the Victor) and invoked the classic phrase 'Victoria Augustorum' (Victory to the Augustii), two very Roman things to find on coins. Theodebert was trying to cement his own authority through Roman imageries and he most definitely was not just an uncouth 'barbarian'. From certain perspectives, Theodebert was very Roman indeed.

As I said, the same goes for most rulers in this period. This was perhaps most evident in Ostrogothic Italy under Theodoric the Great (493-526). As with others, he governed as a Roman ruler, but he was unique in being seen as almost a Roman emperor or even seen as a proper emperor of a renewed Western Roman Empire by some sources:

He so governed two races at the same time, Romans and Goths, that although he himself was of the Arian sect, he nevertheless made no assault on the Catholic religion; he gave games in the circus and the amphitheatre, so that even by the Romans he was called a Trajan or a Valentinian, whose times he took as a model.

His rule was even admired by his enemies:

And though he did not claim the right to assume either the garb or the name of emperor of the Romans, but was called "rex" to the end of his life (for thus the barbarians are accustomed to call their leaders), still, in governing his own subjects, he invested himself with all the qualities which appropriately belong to one who is by birth an emperor. [...] And although in name Theoderic was a usurper, yet in fact he was as truly an emperor as any who have distinguished themselves in this office from the beginning; and love for him among both Goths and Italians grew to be great, and that too contrary to the ordinary habits of men.

Ostrogothic Italy was also immensely powerful during his reign, as Theodoric managed to become the overlord of Visigothic Spain, Vandal North Africa and a smattering of smaller kingdoms to the north, so claims of his imperial rule were actually very tangible. Theodoric had also been a consul from his involvement in Constantinopolitan politics and his heir, Eutharic, was proclaimed to be the 'son-in-arms' of Emperor Justin - a telling statement when Justin would later refuse to adopt a son of the Persian shah. Far from heralding the end of Roman civilisation, 'barbarian' rule in Italy actually ended up preserving a lot of things, to the extent that quite a few historians would now see Justinian's invasion, an Eastern Roman attempt to seize back Italy, as the deathknell of Roman Italy, as the ensuing decades-long war ruined the economy and weakened its defences, allowing the Lombards to invade in the 560s. The Kingdom of the Lombards, which has already been mentioned in this thread, is also relevant to my point here. When it was created in the late sixth century it was probably one of the least Romanised states in the former Western Roman Empire, as it was born from fighting Eastern Roman forces then based in Italy, so avoiding the adoption of Roman institutions/cultural norms was actually more common and a sensible strategy. Over time this changed: some Lombard kings were Romanophiles, whilst more generally the Roman identity of the local population was gradually replaced by a Lombard identity. Was the kingdom becoming more 'civilised' or was the population becoming more 'barbarian'? I don't think we are in a position to judge exactly how this changed, certainly not enough to be able to decide which was superior.

Another thing I should note here is that 476, the traditional date for the fall of the Roman Empire, really didn't mean that much. Most of the Western Roman Empire's territories had already been lost by then and the eventual coup that 'ended' the empire was an internal revolt led by a 'barbarian' general serving the empire, who then proceeded to rule as a viceroy of Constantinople. 476 only became a particularly dramatic year in the 520s, when Constantinople began to consider attacking Ostrogothic Italy - what better way to cement its claims than to say that Italy had long been under 'barbarian' sway and that there was a dire need to restore the empire? In reality, the Western Roman Empire ended with a whimper, not a bang, and civilisation did not fall, but instead adapted to the changed political circumstances of the fifth century.

I might rephrase this question to avoid saying "civilized"--after the fall of Rome, who was still doing the stuff Rome did?

Who was still minting coins, thus indicating a developed cash economy?

Where were the most written documents being produced, indicating a high degree of literacy?

Who, if anyone, had a standing army? Did the Byzantines continue using legions? Did the Persians ever have a professional army?

Who was maintaining roads and public works?

I'd be especially interested in hearing about how civilizations in India compared on these points--I remember reading that in Tamil Nadu there was a lot of cash trade with the Romans and they minted their own coins, and had a relatively sophisticated bureaucracy, but I don't know enough about them to make any sort of educated comparison.

Regarded by whom? I assume you mean regarded by the people of that time, rather than who modern historians regard. And even then, think about how much people would know about a civilization on the other side of the planet? Persians would certainly have a better grip of Chinese and Indian nations versus the Ostrogoths. Visigoths would have a better understanding of North African kingdoms over Turkic tribes.

In northern India around the time of the fall of Western Rome (Byzantine Empire is actually still Rome and they didn't fall until 1453, but I assume you mean Western Rome's fall in 476), northern India was ruled by the Gupta dynasty, considered a golden age. The government was restructured, literacy increased, new technologies were developed, arts and culture flourished etc. But how much would someone in Wales would even be aware of that?

Furthermore, because of trade along the Silk Road, there was certainly a notion of the fact that China was enormous and had more people than Europe could fathom. The largest at the time were either Constantinople or various Chinese cities. But, while the world was at least somewhat aware of how large China was, how much of an idea did they have of their cultural or economic state? If you were to ask this question to someone back then, would China be at the forefront of their mind?