I mean it's not like it wasn't interconnected already. There were extensive trade links between the Middle East and Europe even after the former went Muslim, with Middle Eastern architecture, fashion, cuisines, etc. having a huge influence on Europe at the time. Do you mean that the Middle East is seen as less of an "other" by Europe? That could be possible, but there will still be significant differences. While the Eastern Roman Emperors definitely tried to create a sense of loyalty of those regions to the empire by bringing their Churches into communion with Constantinople, many Christians in the region, especially in places like Egypt, split with what would eventually become Catholicism and Eastern Orthodoxy in the 400s and would eventually be seen as heretical by Christians in Europe. There's also Persia to consider, and whatever steppe cultures like the Turks and Mongols come out of Central Asia. If any of those groups can conquer the Levant for a significant period of time they will pull it further out of the orbit of Europe.
I think without Islam what you're likely to see is a Middle East Levant that doesn't stay under East Roman domination as the empire weakens, and that is likely to be seen as "other" in the same way Catholic Europe viewed the Eastern Orthodox states like Byzantium and Russia. Christian, but not the right kind of Christian and thus not fully "us."