Arabia would not extend past the Syrian Desert. The Levant, Mesopotamia and Egypt would not bee... Arabic...
I'm not entirely convinced of this. Arabia didn't need Islam to unify. It helped, sure - but you already had Arab migrations into Mesopotamia at Al-Hira, the Annizah, Nabateans, etc. The region wasn't particulary stable on the Roman or Persian side.
Just to look at the Persians, where the most opportunity lies, lets get back to Al-Hira. Still protected by the Lakhmids who'd been punished by the Persians a couple of decades earlier. With the instability following the execution of Khosrau II. This actually leaves the Lakhmids in an interesting position, they could both avenge their previous punishment, and become very powerful if they basically open their doors to their fellow Arabs, especially Arab Christians and even local Christians. You could see the rise of a Christian Arab Mesopotamia quite easily. Even if it isn't independent, it could rapidly establish itself as a power-broker in Persian politics.
As with Syria and Egypt - the Romans can always use soldiers. Migration, settlement, recruitment, repeat - assuming they live on the border fringes, you could still see the region Arabise, especially if the Roman Empire falls apart ITTL. Regardless, they can become a significant demographic within those regions.