This is not true at all. No historian nowadays would see the (Eastern) Roman Empire as something so worthless. Remember that this was a book written about philosophy by someone who was not trained as a historian from 1945, so the quality of historical information it provided is strictly limited. This entire passage appalls me in terms of its understanding of the past:

The barbarian invasion put an end, for six centuries, to the civilization of western Europe. It lingered in Ireland until the Danes destroyed it in the ninth century; before its extinction there it produced one notable figure, Scotus Erigena. In the Eastern Empire, Greek civilization, in a desiccated form, survived, as in a museum, till the fall of Constantinople in 1453, but nothing of importance to the world came out of Constantinople except an artistic tradition and Justinian's Codes of Roman law.

No-one today would equate civilisation only with the Roman Empire and the Renaissance or suggest that the 'barbarian' invasions were so devastating. This simply represented the long-outdated view that 'Dark Age' Europe was a continent of ignorance and blind belief. Instead, historians are now aware of the diversity of thought within the dynamic world of late antiquity and the early middle ages. It was certainly different from the classical world, but was it inferior? We should not forget that the classical world was a brutal place where the elite ruled over a vast underclass that never enjoyed the fruits of 'civilisation' - the philosophy, science and refined table-talk we so associate with the Greeks and Romans during their 'golden ages' were never what the average person on the street experienced. The past was a bitter and brutal place; naturally, the medieval world was no exception.

This view was however dominant for a very long time, even when talking about the Roman Empire/Byzantium, which was often seen as an inferior copy of its 'predecessor'. Judith Herrin's Byzantium: The Surprising Life of a Medieval Empire (2009) summed up this quite well:

Yet the modern stereotype of Byzantium is tyrannical government by effeminate, cowardly men and corrupt eunuchs, obsessed with hollow rituals and endless, complex and incomprehensible bureaucracy. Montesquieu developed these caricatures during the seventeenth century as he tried to explain the reasons for the decline of the Roman Empire, and Voltaire gave them greater prominence, adding his own passionate elevation of reason above religion. While the former dismissed ‘the Greek Empire’, as he called it, because of the excessive power of monks, attention to theological dispute and an absence of the recommended separation of ecclesiastical from secular matters, the latter could condemn it utterly as ‘a disgrace for the human mind’.

This stereotype is of course utterly untrue, as Herrin's book, or indeed any book written by a sane historian from the last century, would provide you with a completely different picture. To quote Herrin again:

As I hope to have shown, far from being passive, Byzantium was active, surprising and creative, as it reworked its prized traditions and heritage. It bequeathed to the world an imperial system of government built upon a trained, civilian administration and tax system; a legal structure based on Roman law; a unique curriculum of secular education that preserved much of classical, pagan learning; orthodox theology, artistic expression and spiritual traditions enshrined in the Greek Church; and coronation and court rituals that had many imitators.

Just from reading the sources, you would not get the impression that they have forgotten their past nor were they intellectually stunted. People doubted the cult of saints,^1 their republican past was valued,^2 men of diverse learning were often in positions of power,^3 and more than few women rose to the top of the imperial hierarchy.^4 This is ignoring the many theologians who made their name in the east, such as Maximus the Confessor and John of Damascus - God mattered to these people and they used their fearsome intellect to elucidate their beliefs; why should they be dismissed in favour of secular philosophy, which in any case continued to be learned and developed within the empire? Above all, the Byzantines were proud to call themselves Romans, something that we should never forget when we talk about Roman history. Without these medieval Romans' efforts in remembering the past, such as the compilations made by Constantine VII Porphyrogennetos or Patriarch Photios of Constantinople, we would know much less about the ancient world.

Even disregarding their culture, which was frankly no more or less impressive than their contemporaries', since theologians and remarkable individuals can also be found amongst the so-called 'barbarians',^5 the Roman Empire was a Mediterranean power until its terminal decline after the thirteenth century. It cannot be ignored when studying the history of Western Europe,, since it regularly intervened in western affairs, such as during Justinian's reconquests in the sixth century, Constans II's personal visit to Rome in the seventh century, and Alexios I Komnenos' call for western aid that precipitated the First Crusade in the eleventh century, to name just a few examples. The Roman Empire continued to be a vitally important part of world history until its very end, so it's always worth learning more about it.

In all honesty, I find listing each culture's contribution to civilisation annoying, since it implies that certain things were better than others and that we can judge each culture's worth. In reality, we can't. I study the Roman Empire because it is interesting, not because I believe it contributed great things to the world. The past is a fascinating place and I feel privileged just to be able to explore it; to go around condemning people who were doing the best they could in trying circumstances, who experienced trials that I never had to endure, seems to me to be utterly unempathetic.

For further reading, Herrin's book is very good, as is Averil Cameron's The Byzantines (2006). I haven't read Jonathan Harris' new book, The Lost World of Byzantium (2015), but it seems to be a decent introductory book as well. Another choice would be Peter Brown's classic The World of Late Antiquity (1971), which was perhaps the first book in English to explore how dynamic late antiquity was and to argue that it was something to be studied in its own right, rather than being seen as the tail-end of a 'superior' antiquity studied by classicists.

---

References:

1. M. Dal Santo, Debating the Saints' Cults in the Age of Gregory the Great (2012).
2. A. Kaldellis, The Byzantine Republic: People and Power in New Rome (2015).
3. Michael Psellos from the eleventh century is perhaps the best example.
4. Empresses Theodora and Sophia from the sixth century are two great examples, Anna Komnene from the twelfth century meanwhile provided ample evidence of her literary education through her Alexiad, a history of her father's reign.
5. Isidore of Seville, the Venerable Bede, and Aldhelm of Malmesbury come to mind.

Follow up question: given that Russell is a philosopher and his primary interest lies there, what substantial philosophical contributions did Constantinople give to the world?

Not knowledgable about philosophy at all, but just had a thought he might have meant it more in this sense rather than the general one.

It should be noted that Russell's book is generally badly received by philosophers. He makes huge sweeping claims about philosophers with little argument to back it up: for example, in the same book he directly links certain philosophers with Nazism- not an absurd claim to make as there was a tendency at the time to do so (Karl Popper links Plato to Nazism, Isiah Berlin links Hegel etc) but Russell seems to do so almost arbitrarily. Although I know nothing about Byzantine philosophy, I would be inclined take his assertion with a pinch of salt.