In short, not really. And this is for two main reasons, first the "Great Schism" wasn't all that big of a deal in the immediate aftermath that forced various churches to fall into various camps, and second the ability of either Rome or Constantinople to effectively sway the churches in disparate parts of Europe was extremely limited, so other concerns such as geographic proximity outweighed theological or political disputes.

I do want to unpack that first point though, as the "Great Schism" that split the Christian world is an extremely misunderstood world event. The disputes between the Latin Church and Greek Church were not new in the 11th century. Disagreements over the importance of the Papacy, the unending tension over the presence of "filioque" in the Latin translation of the Nicene Creed, political disputes between the (Holy/German) Western Emperors and the (Greek/Byzantine) Emperors of the East, and more all contributed to fragmenting Church authority. Indeed look at Italy for an interesting case, where geographic proximity placed the Papacy right next to many churches that still looked to Constantinople for guidance until the Norman conquests of the two Sicilies in the 11th century. Relations between the two Sees, Rome and Constantinople, were perpetually low. One particular low point led to the "Great Schism" but there is actually little evidence that this situation was treated with enormous gravity at the time. It was instead seen as simply another nadir in East/West relations that would improve with time, and indeed this is exactly what happened. The schism did not prevent three separate Greek Emperors, Alexius, John, and Michael Commenus, from seeking aid and support from the Papacy (and winding up with the crusades instead) or from the Byzantine court continuing to build relations with Catholic parts of Europe. Indeed not even the 4th Crusade was able to firmly destroy these western centric tendencies among the Byzantine elite, as even up until 1453 the Byzantines courted the Papacy and western powers as potential allies against the encroachment of Turkish powers such as the Ottomans.

Particularly in Eastern Europe the situation was much less strictly defined than modern perceptions of the "Great Schism" may indicate. Indeed, some scholars such as George Demacopoulos have argued that the crystallization of "Catholic" and "Orthodox" didn't happen until the 4th Crusade and that in the time before that, the Latins in the East and Greeks in the West were still able to effectively participate in and navigate through the various churches that ostensibly belonged to the "other" church. Greek Christians and Latin Christians still recognized the legitimacy of the other up until the 4th crusade and the fallout of the crusaders' seizure of Constantinople.

The second issue is one of both geographic proximity as well as political expediency. By the time of the "Great Schism" the Greek Church was in dire straits. The major centers of Eastern Christianity had been cut off from Constantinople in some cases for centuries by this point, with Antioch being the last major city of the Roman East to fall to Islamic forced (before being recovered, lost again, captured by crusaders, captured again and so on). Churches in Jerusalem, Alexandria, the Caucasus and more had already been lost to East Roman oversight (with a handful of "Melkite" churches being the exception in Jerusalem and the Levant). In the west a similar situation had also developed. As a result of Byzantine weakness, the Normans, invited by the Papacy, conquered much of Byzantium's territory in Italy and even launched invasions of the Balkans, sundering the most sympathetic churches in western Europe from Byzantine control.

In the other corners of the Latin Church world, there was simply no reason to adhere to the "Orthodox" Church's authority as Rome was the much more important center of the church. This was due to geographic proximity yes, but also cultural importance, all of western Europe (minus some parts of Italy) looked to Rome for spiritual guidance and importance, as well as political support. It was Papal support that earned the Carolingians their throne, sent missionaries to Britain, invited Normans to conquer the Mediterranean (and England). It made no sense for any Western European power to break with Rome in favor of a distant and, at the time, reeling Greek Church. A church for whom the majority of Western Christians had little more than contempt for. The Greek Church and Empire were the targets of derision and mockery by western Europeans. They were seen as effeminate, untrustworthy, cowardly, and self serving (for their part the Greeks viewed the "Franks", as they called them, as uneducated and uncultured bumpkins whose only virtue was their battlefield bravery).

Now there were cases of some churches around the Christian world who were tied to Constantinople over Rome, but these did not end up in that camp because of the Great Schism, they already had relations to Constantinople beforehand. These were the Orthodox states in Eastern Europe such as Bulgaria (when it existed as an independent state which wasn't often), the Russian principalities, and Balkan powers such as Serbia. Other states that had seen conflict (mostly just words and theologions bickering) between Rome and Constantinople, such as modern day Hungary, Czechia, Slovakia, and other central Europe states had already entered into the sphere of influence of Rome and had no reason to break with Rome over the schism.