In short, yes but for reasons quite different to what you might think.

By 750, over a hundred years after the death of the prophet, the borders between the Orthodox Christian world and the Byzantine world were roughly the same as they would be until the arrival of the Turks around the turn of the millennium. In these years, Islam was not a prosletysing religion, meaning that it was not a religion which actively encouraged people to convert, instead preferring to rule large Christian and Jewish populations they hoped would eventually convert through the jizya tax (muslims were exempt from it) and education. Therefore, Muslim ideas did not spread beyond the frontier established between Byzantium and the Caliphate to any great extent. This was no doubt helped by the fact that emperors after the Muslim conquest became much more aggressive in their Christianity, and Jews and non Chalcedonian Christians were heavily persecuted.

After 1091, where much of the Eastern part of the empire was lost to the Turks, again, few Muslims would have crossed the redrawn frontiers.

The only point therefore, when you can say that a Byzantine Emperor of Constantinople ruled over a significant Muslim population is during the various territory swaps which occurred when Bey Orhan (1326-1362) of the Ottoman Emirate (later empire) had Byzantine Emperor John VI as a tributary. After the Byzantine civil war, the emperor was allowed by his Ottoman overlord small control over parts of Thrace, the immediate area around Constantinople. By this point, Turkish immigration to the area would have meant that there were some, although still not many when compared to the Turkish lands, in these areas remitted to the emperor. John VI's successors proceeded to lose said lands quite quickly and until the conquest of Constantinople in 1453, the so-called 'empire' was effectively a rump state based entirely in Constantinople and entirely subject to whims of the Ottomans.

So really, the Byzantines never ruled over large groups of Muslims in the form of Empire. For most of the history between Byzantium and the Muslim states of the Middle East, it was one way traffic, the Byzantines never recovered their vast lost territories in the Middle East, and after 1091, were neither able to make gains in Anatolia. The Muslim conquests would actually have made Byzantium, in a way, more hardline Christian. Many of the softer Christians or Christians who didn't follow the strict Byzantine line were quite happy to enjoy the religious freedom of living under medieval Muslim rulers (how times change!) and would have stayed put after Muslim conquest, whilst the traditionalists fled to the West or into Constantinople itself. This led to a society, which by the end, was pretty intolerant, and wouldn't have accepted a Muslim minority within its borders whilst facing existential threats from Muslims outside.