The Eastern Empire in its early years would have been extremely difficult to unify. Not only are there diverse cultures and peoples, the geography ensures that the Byzantines would have had to keep a standing navy in prime shape at all times in order to link Africa with Greece, which it obviously failed to do.
If Islam had not risen, I supposed the Byzantines would have had to continue their power struggle with some other Middle-Eastern empire (which would certainly have arisen in the place of Persia), or vie with Western Europe. Either way, the empire being located at the borderline of Europe/Asia would not have lasted much longer than it did.
My personal feeling is that Islam kept Western/Eastern Europe from going at each other's throats immediately by presenting itself as a unified danger. The religious differences between Orthodoxy and Catholicism would have easily been enough to start war had not the early Arabic states kept them in an uneasy alliance.