When the Western Roman empire fell to the Gothic king Odoacer when he and his troops sacked Rome, the Eastern Roman Empire already belonged to a separate emperor (then Leo). This changed took place as early as Diocletian, but after Constantine (who ruled a united empire), the western and eastern halves remained separate, allowing the east to endure until the Ottomans sacked the city in 1453.
The residents Eastern Roman Empire were Greek-speaking Latins, and Christianity is largely responsible for the maintenance of a society that can rightly be termed "Europe."
First, the Church is responsible for overseeing many typical institutions that we would call necessary for an urban society (not only churches, but also hospitals, orphanages, and schools). This urban infrastructure allowed the Byzantine empire to develop and flourish, and thus arm itself against attacks later from the Muslims and Turks.
Next, the Church is responsible for the various monasteries that were responsible for translating and transmitting classical texts. While religious figures paid various levels of attention to (secular) classical texts, churches generally maintained monastic libraries that served as a repository for classical texts that came to influence the West after the Middle Ages. Some classical texts were transmitted accidentally, by virtue of being found on the back of the papyrus used to record Scripture or other general monastic writings.
Finally, the Church was sanctioned as the official religious of the Byzantine empire, and thus enjoyed formal ecumenical councils that we now recognize as hallmarks of developed bureaucracy (religious or otherwise). The Church also required prominent architecture, and (despite a brief period known as "iconoclasm" in which icons were forbidden) many churches that have now been converted to mosques remain standing today in the Byzantine Empire's capital, Constantinople. The most famous of these is the Hagia Sophia, originally built under Constantius II.