This is an enormously complex and interesting question which is beyond my power to fully answer, but I’ll give it a go and hope it helps.

It depends what you mean by Europe. I will assume you mean those areas of Western Europe which were occupied by the Roman Empire

Firstly, I would say to clarify that Iberia, then known as Al Andalus, was united in the Umayyad Caliphate and Italy was, for a time at least, united with the Eastern Roman (Byzantine) Empire. Secondly, the Carolingian Empire under Charlemagne did in-fact unify large areas of Western Europe in the 9th century and would eventually become the Holy Roman Empire which would last until the 19th Century.

You could say that Europe was united ideologically, if not politically, under Catholicism with every state recognising the supremacy of the Holy Roman Empire.

However it is true that while Europe languished in the dark ages, the Umayyad and Tang enjoyed golden ages.

Western Europe was the least developed area in the Roman Empire with low levels of trade and urbanisation. The most valuable Roman territory was in the Middle East which was taken by the Caliphate.

Also The Caliphate was ruled by a single group: the Arab Muslims, whereas the various Germanic tribes which occupied the former Western Empire were independent and distinct from one another.

Because the Muslim Caliphate was hostile to Christian Kingdoms, Europe was blockaded, and thrown back on its own meagre resources. For example the importation of papyrus from Egypt ceased which caused the price of paper, to rise sharply and literacy levels to drop accordingly. The presence of a strong literate class is essential to build any sophisticated state. This is known as the Pirenne Thesis and is somewhat controversial. In contrast to Europe China is very self-sufficient economically, and the Umayyad’s had access to trade routes spanning the known world. Europe is much more segmented geographically than either the Middle East or China. Bodies of water like the Rhine and English Channel and mountains like the Pyrenees and Alps form natural boundaries which favour the defender, so any would-be empire builder has to struggle with the terrain.

European kingdoms were also plagued with Vikings from the 8th to 11th Centuries which hampered their attempts to strengthen their administrations.

Some previous related questions:

In the 5th century both the Roman Empire and China were invaded by nomadic peoples and were divided. But why did the idea of unified China survive while Europe developed into many different states that never unified?

Why have the histories of China, the Middle East and Iran seen more imperial unification compared to the history of Europe? How come China would fall apart and reunite, while Europe (Rome) fell apart and fragmented permenantly?

Why was Europe never unified the way that China was?

Why was China able to eventually re-assemble into a single polity after the collapse of the Han dynasty, but Western Europe was not after the fall of the Western Roman Empire?

How was China able to not break up into small, European style countries?

Why did China recover more quickly from the fall of the Han Dynasty than the Mediterranean did from the fall of the Roman Empire?

Somewhat less related questions:

If China used to be several distinct nations, is the historical difference between the nations that unified into China greater than any neighbouring countries?

Is it fair to say that "Chinese civilization" is more "continuous" than "Western civilization"?

Why are the people of a land as big as China relatively so culturally similar while Europe and the Middle East are so full of different people and cultures?

China is huge. How did it become just one nation?

Why is Chinese civilization considered to be continuous, even though China has been invaded, conquered, and divided numerous times?

Why do we have so few really old countries in Europe?

Maybe add to FAQ?