It seems to me that the Great Schism is still going to be on its way in some alt-form or another, and with Arabia much closer to Constantinople than Rome, wouldn't that put it in the *Orthodox camp?
Unless, of course, lack of pressure from Islamic empires means the ERE is able to reassert its authority over the west with greater success, thusly aborting the power of the bishop of Rome and uniting the Church under the emperor... But then, that wouldn't really be "Catholic" (at least not as we know it) either.
Hmm... Tricky.