Europe prays really, REALLY hard.
But seriously, if the Arabs overwhelm Constantinople and get into the Balkans, then isn't Western Christiandom doomed? Islam will convert the Rus, the Norse, and completely encircle "The West" (British Isles, France-Germany, Italy, and Spain).
Now I think that the flight of thousands (maybe millions) of East Romans to the west will have some serious knock-on effects, so perhaps some kind of stable political system (ie not the post-Charlemagne fratricide) will come into being. More centralized, less chaotic. So maybe this much smaller West can weather the Arab-Islamic storm, beating off the killing blow until the Arabs splinter at home and take the pressure off.
Off the top of my head I would say that some kind of more centralized Europe of this era will have to rely heavily on the Church, so I would expect a much more powerful church, along with a much expanded Empire. The East Romans will bring their regalia with them when they flee Constantinople, so the "Roman Emperor"- probably whomever the strongest Frankish lord is at that particular point is, gets that title.
With the East Romans' much more higher level of administrative know-how I would expect that they would quickly rise through the Church and Imperial hierarchy. They also might bring that later "Baliseus" (can't spell) idea with them, with the Emperor as the state, that is having unlimited power.