I'm going to have to disagree with /u/waterplace. In the centuries before the rise of Islam, Christianity was almost certainly the majority religion in the middle east with not just Byzantine cities but also Sassanian mesopotamia being overwhelmingly Christian at the very least from around the 4th century through to the 8th. See, for example, The World of Late Antiquity by Peter Brown.

The question, however, is a difficult one to answer as there was no real census data being taken. One method, taken by Richard Bulliet in "Conversion to Islam in the medieval period" examines when names start to become Islamic in biographical dictionaries as well as when certain regions become more prominent, implying that they had converted earlier. By this method Iran (which in his methodology excludes mesopotamia) probably converted quickly, and was majority Islamic by the 9th century. As late as 750, however, less than 10% of the population were muslims. In Iraq/Mesopotamia, which had a large settler community, the conversion was earlier still, with a majority of Islamic names by the end of the 8th century.

Egyt, Syria, and Tunisia have problems with sourcing (the biographies have fewer recorded names) but probably were majority Muslim by the end of the 10th century.

Reading his work it becomes quickly apparent that it is filled with assumptions and outright speculation. But in the absence of better source material it's the most reasonable method I've come across.

source: CONVERSION TO ISLAM IN THE MEDIEVAL PERIOD: AN ESSAY IN QUANTITATIVE HISTORY Bulliet, Richard W., 1979.