The best answer may well be some time in the 1500s. Recall that as the era shifts from the late Classical to early Medieval, all of these people are still speaking the Roman language (Vulgar Latin, which evolves into the various Romance languages), following the Roman religion (Christianity), obeying Roman legal codes, and in many aspects, still following Roman customs and experience the same Roman economic and administrative system. In the case of the Byzantines, there is continuous institutional survival until the Fourth Crusade in 1204. Now, what that Roman society looks like in the 500s is very much not the same Roman society we conceptualize of in the 100s, but there is largely no clean break [1]. There is political disunity, but that doesn't necessarily mean that there isn't a recognized commonality of culture (cf. China, where similar major periods of disunity are still categorized as all being part of China).
The major rupture is the Protestant Reformation, where the split between Protestant and Catholic Christianity proves irreconcilable, and results in the end of the notional idea of a unified Christendom. This is also when you start to see an end towards the practice of writing in the literate language of Christendom (i.e., Latin) and instead move towards working in the vernacular, especially in endeavors like scientific research.
[1] The major exception is Britain, where the end of Roman rule is very abruptly realized, and there is a distinct clear horizon between sub-Roman Britannia and Anglo-Saxon Britain. But the British experience is largely the exception, not the rule.