Perhaps, but this did not occur to any significant amount till after 1400. By that time, a lot of Greek learning had made its way into Western Europe thru Spain and other Moslem conduits. I contend that this earlier Greek learning was far more influential in the history (even the formation) of Western civilization than some humanistic polishing acquired during the Renaissance.
Without the Greek modes of thought transmitted by the Moslems, it is unlikely there would have been a High Middle Ages or a Renaissance.
Again, I believe that Moslems are being given far too much credit for the transmission of Greek thought.
To say that cultural exchange between Constantinople and the West "did not occur to any significant amount till after 1400" erases the Fourth Crusade and the Latin Empire of Constantinople from the pages of history.
In 1203-1204, the Doge of Venice persuaded the Fourth Crusaders to attack Venice's maritime rival, Constantinople, as payment for Venetian sea transport of the Crusaders. Constantinople was conquered by the Fourth Crusaders in 1204.
From 1204 to 1261, the Catholic West ruled Constantinople as the Latin Empire of Constantinople.
The greatest conduit of Greek thought and Greek treasures into the Europe of the Middle Ages was the Most Serene Republic of Venice and not the Arab world. The Latin conquest of Constantinople in 1204 wasn't nice and it wasn't pretty but it did give the West, principally Venice, a 57 year long rule over the direct descendant of the Eastern Roman Empire.
This is fair, since Venice itself was a former province of the Byzantine Empire. The ruler of Venice was the Doge, which is Venetian dialect for Duces, a Greek title for Byzantine noblemen. The Aristocratic title 'Duke' in the west derives from the same source. Late in the first milennium the Byzantine province centering on Venice became increasingly independent from the Imperial capital, partially due to its support for Pepin and Charlemagne of the Franks, against the Lombard rulers of north Italy, recognized by Byzantium. The relationship with the Imperial capital waxed and waned over the next several centuries, with the vagaries of war and politics.
Even now there are churches of Orthodox architecture within this area of Italy and nearby Croatia. The great Saint Mark's Cathedral in Venice is an example, and still has icons of the Emperor Justinian and his Empress Theodora, which are great tourist attractions. Venice took control of the coastal area of Croatia, including the islands, nearly as far south as Greece. This is the source of the Roman Catholic influence in the Balkans, since Venice stayed with the Pope in Rome instead of the Ecumenical Patriarch at Byzantium after the Great Schism of 1054 AD.
In any case the Byzantine/Venetian/Italian connection maintained a close relationship of the West with the continued and weakend Roman Empire in its Byzantine capital well into the Renaissance period. As Byzantium fell into decay, and early modern Italy grew wealthy, Byzantine scholars and artists were hired to bringt the knowledge of the ancient world to Italy, and on to France and the Holy Roman Empire of the German Nation.