>The current muslim countries actually have a lot of Roman influence
Like what? All I can think of is anti-roman influence. It's very hard to settle that in a short answer but I think Islam is almost the opossite of hellenism and "roman influence".
>First of all, the Eastern Empire had dealings with Muslims 1000 years after the west fell
The same dealings the Western Empire had with the Huns. We are not calling the Huns as heirs of the romans right? It was a hostile relationship from day 1.
> Sure, muslim countries are not children of the Roman Empire. The empire was, pretty much but not only, a container for hellenized mediterraneans, kind of what Islam rejects.
That is bullshit. The current muslim countries actually have a lot of Roman influence. First of all, the Eastern Empire had dealings with Muslims 1000 years after the west fell. Caliphate took over many of the Roman institutions when they took over North Africa and the Levant.
You can not compare current Islamists with Islam at that time.