This has been one of the more enlightening discussions on this board.
The problem with early PODs is that you really can't see much past the next century due to butterflies, if only due to different people being born. But I'm convinced by the above commentators that without Islam, the Arabs would have expanded due to overpopulation and the weakness of the Eastern Roman and the Sassanid empires. But the Arabs would not have been united and would have less motivation, so they would not have gotten that far.
In particular, while the Sassinids fell without much of a struggle, Heraclius put together an East Roman counter-attack and it was touch and go for awhile. Reinforcements had to be moved from the armies in Mesopotamia to Syria, that if the expansion was done by disunited Arab tribes would not have been possible. And even the Iranians retreated to the Iranian plateau at one point, and that might have worked with a less forceful Arab advance, the Arabs would never have broken into the Arab plateau.
Egypt fell to the Arabs with almost no fighting, though again here the East Romans managed a counter-attack, but if the East Romans can hang on to Palestine, that blocks any moves into Egypt. I can see the East Romans hanging on to a coastal strip similar to what the Crusaders had. This implies they would have eventually lost Damascus to the Arabs. The Arabs would have overrun Mesopotamia except the East Roman part.
That gives two Arab kingdoms, one in Mesopotamia and one around Damascus. The Ummayyads of OTL would likely wind up ruling the Damascus kingdom. I'm not sure which clan would have wound up in Mesopotamia. Both would have raided vigorously. Eventually the Syrian Arabs would have converted to Monophysite Christianity, and the Mesopotamians to Nestorian Christianity, as that was the faith of the majority of the population in those areas. This happens to be very analogous to the Vikings, lots of raids, some small states in the raided areas, followed by conversion to the religion of the region.
Past the 6th century its difficult to see, but you start getting changes in the situation with Justinian II.
This has major butterflies starting with no Arab expansion into the Maghreb, Iberia, Central Asia/ Afghanistan, or the Sind, though there is a chance the Arabs still reach the Sind by sea and found the IOTL kingdoms. Mediterranean trade continues as before. There is no or less of a sharply defined break between the civilization of Western Europe and the civilization of the Near East. Monophysitism remains a huge problem for the Emperors in Constantinople. However, the iconoclasm controversy is butterflied away.
The East Romans still have their hands full defending against the Arabs,they just get to do it farther forward, so this doesn't help them in the Balkans or Italy. They do get to keep Sicily and North Africa, at least for longer than they did IOTL.
I agree that keeping a Christian Near East, with a patriarchate in Alexandria as important as the ones in Constantinople and Rome (Antioch and Jerusalem were second string, and any Nestorian patriarchite would be on the fringes of the Christian world), would have effects on the internal development of Christianity. Assuming the Iranians didn't just convert to Nestorian Christianity themselves, which is unlikely, Zoroastrianism would have developed more into a popular religion and maybe would have spread to India as much as Islam later did, with Nestorian Christianity making more inroads in the south. Central Asia would have remained mostly Buddhist, and either Catholic or Orthodox missionaries would have converted the Berbers and the residents of the Sahel.