What if the Roman Empire had conquered the whole Arabian peninsula? Could they romanize the Arabs? And would that change history?
Well, I think they can conquer at least more of the Pennisula. The entire I don't know but Mecca could be possible. A better sea trade there would also help a lot keeping the provinces there. I do wonder however if the Romans would want to have it - most of it is desert after all and pretty useless, and it is a large area for an army to occupy. I think they might be Romanized, but I think Hellenization is more likely - this also happened with the Nabateans OTL.
If Arab raiders were attacking the Roman provinces and Roman caravans a punitive expedition could follow with the aim to subjugate them. This could lead to the conquest of Mecca. An insurrection in Mecca would lead to the destruction of the town and foundation of a roman colonia in its place with a temple of Jupiter in place of the Kaaba.
This. The only area of Arabia worth something to the Romans at this time was modern day Yemen, and there is a huge area of desert/sea to go through to get to there. I'd expect that the Romans would be more interested in Mesopotamia.The Arab provinces would be absurdly poor compared to the eastern provinces and I don't see much interest in them.