It is not a very plausible scenario: as others have pointed out, there is little reason to assume that the Caliphate can conquer Constantinople and Greece and Africa and all that, but doesn't make any inroads into Sassanid Mesopotamia, which is right next door to Arabia and a source of wealth.
But if we assume it went that way, I still don't think an early caliphate would really call themselves the new Roman Empire. Yet, that doesn't mean they wouldn't absorb a lot of different influences from OTL.
As far as conversions are concerned, I see massive conversions in the European territories, too. There is no "European cultural resistance" to Islam, that's ontologising an interpretation of historical developments.
This would be a Mediterranean-centered Islamic world, even more so than IOTL, which means Italy is next at some point, and I don't see why the rest of Europe wouldn't ultimately also convert. (If Europe still ends up dominating the world for a couple of recent centuries, we'd be sitting here and debating if that had soemthing to do with its Islamic nature...) For the time we call the Middle Ages, though, that means massive changes everywhere. India has already been mentioned - other important areas are Central Asia and Africa, where Euro-Islamic powers would contend with Iranian, Indian etc. contenders. IOTL, the Indian Ocean became a Muslim lake, over time. ITTL, this Muslim lake is the Mediterranean.