In the great age of French culture, the cultural influence of French culture was very great and widespread in Europe.
In the great age of Spanish culture, the cultural influence of Spanish culture was very great and widespread in Europe.
In the great age of Muslim Spain centuries earlier, it was a source of knowledge about ancient Greek culture, with western scholars translating Arab translations of Greek texts into Latin. Western scholars also learned of the advances of Arab culture in Spain and other border countries, advances like Arabic numbers, the numeral zero and algebra, for example.
In the comparatively dark ages of western culture in the worst periods of the Middle Ages, people drew inspiration from the memory of the lost glories of the ancient world, as well as from the one Christian society where those ancient glories still lingered - what was left of the Roman Empire, which was usually the strongest military, political, diplomatic, and economic power in the two overlapping regions of Europe and Christendom, and was by far the most scientifically and culturally advanced society.
So for many centuries the cultural influence of the society based in Eastern Europe and Asia Minor on Western Europe was very strong.
Most ancient Greek texts are known mostly from translations into Arabic in Abbasid Baghdad that were later translated into Western languages, or from copies made in the medieval Roman Empire in eastern Europe. Many of those medieval copies were taken to western Europe about the time of the fall of the eastern empire, helping to fuel the Renaissance. And many lost works of ancient Greek literature and mythology are known only by Byzantine discussions of those lost works.
And of course the military strength of the remnant of the Roman Empire, which fluctuated vastly over time, was the chief barrier between Eastern Europe, Central Europe, and Western Europe on one hand, and powerful invaders from the east.
Western Europeans often celebrate the victory over Muslim raiders from Spain at the Battle of Tours in 732. But the Muslim expansion began in Arabia and was directed from the capital of Damascus in Syria from 665 to 750, and then from Iraq until the the Caliphate dissolved into independent states. And the direct route to Europe from the Muslim centers in Syria and Iraq went through the eastern Roman empire and was effectively blocked by it.
So the cultural influence of Eastern Europe on Western Europe was vast. Without the military power of the Eastern Roman Empire, Western Europe would quite likely be a group of Muslim countries. If the Roman Empire had remained as powerful as it was in 1025 forever, it would have protected Western Europe from eastern invasions forever.
But when the remnant of the Roman Empire crumbled and lost its military power, thanks in part to attacks from Western Europe, the Ottoman Turks emerged to fill the power vacuum and eventually conquered a fifth of all Europe. And threatened the rest of Europe.
So I would say that there was considerable cultural influence from Eastern Europe on Western Europe.