If the Muslims did take Constantinople either in the 600s or 700s AD, how much further in the Balkans would they have gone?
HG Wells speculated that had the Muslims concentrated on Constantinople and won, nothing after Constantinople would have stopped them, and the pagan Slavs, Avars, Bulgars and maybe even Saxons would have converted to Islam.
But that touches on a problem with conversions of these tribal peoples and empires to Islam, that in Omayyad and Abbasid times non-Muslim rulers could not just "convert" their domains. Conversions were only considered genuine if they subjected themselves to Caliphal political rule, at least that's how this poster described it:
John7755 يوحنا John7755 يوحنا is offline
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I doubt that the Khazars would convert to Islam within the timeframe, mainly do to the facts of Abbasid influence and would be more likely to convert to Orthodox Christianity. At this time the Arab caliphates (Abbasid and Umayyad) would claim sovereignty over all believers and would fight to force the said believer to join the caliphate, it was only till the Turks cane on the scene plus enormous social upheaval (Shia and Khawarij revolts) did the Caliph stop this policy. At the same time Byzantium did not claim such power and attempt to gain it, this is shown through the example of Russia, which would be an example of how Khazaria would look.
So in short, the Khazara wished to keep themselves independent from both polities (hince conversion to Judaism) but if he has to choose they would either stay Tengri or would have a Russian style conversion to Christianity.
So this would seem to indicate that, at least until the Abbasid decline (late 800s, or 900s or 1000s) even an Islamic empire that had taken Constantinople limited its prospects for converting large territories to the north of the Mediterranean, because self-respecting Avar, Bulgar, Magyar and Rus khagans, would be opposed to converting as it meant giving up their temporal authority.
So then the Muslims after Constantinople would only convert populations that they literally conquered themselves. So effectively this might mean southern Thrace and and the Aegean littoral, with people's to the north clinging to paganism, or still becoming Christian (or perhaps going for the Khazar Judaism approach) for a lengthy interval before local rulers would feel comfortable converting to Islam.
Excluding conversions of the states set up by the steppe or norse nomads, how much of the Balkans would the Arab Caliphate be able to conquer and how much would it even be able to conquer. Are the Danube, Morava and Drina rivers actually a stretch?