The existence of the Vatican state today, and that of the Holy Roman Empire in Medieval times, I think proves you wrong.

No, I don't think they do. Vatican is a state only in the name, and as the famous quip about the Holy Roman Empire goes, it was neither Holy, nor Roman, nor an Empire :-)

The relationship between Christianity and state power is a complicated one and not reducible to a yes/no question.

I also think that if you replace the wiggly "impose itself in law-making and state politics" with much more direct "seek to rule, that is, solely control the state power", you'll find that Islam and Christianity are quite different in that respect. The notion that Islam and Christianity have essentially same attitude towards state is nonsense.