No they didn't. The Holy Roman Empire didn't even exist when Arab Muslims conquered the lands from the
Byzantine Empire (aka the East Roman Empire).
Those lands didn't belong to the Byzantine Empire anymore than they did to the subsequent Muslim conquerors. Most of the territory of the preceding Roman Empire was acquired by conquest as well. The areas held by it and its successor Byzantine Empire which the Muslims conquered were held for only a little over a century longer than the Sultanates approximate four century reign over the same territory up until the time of the First Crusade. With the exception of Gaul (Spain), none of the land was in Europe and most of it was in Asia Minor. Gaul was taken from its Muslim rulers by the Reconquista, not the Crusades.
The Crusades were a series of invasions into Asia Minor long after the Byzantine Empire lost the territories to the Muslims ending over four centuries of peace, trade and commerce between the Sultanate and the Empire . Your attempt to mischaracterize the nature of the Crusades as anything but a campaign of conquest and religious intolerance is nothing more than fabrication.
The Arab Muslims that ruled over Asia Minor at the time weren't radical Muslims. Much of the time the area was ruled with great tolerance (certainly greater tolerance than the Christians behind the Crusades). Christians were allowed to conduct pilgrimages to various holy sites and live and openly worship within the Sultanate. It a was a time of peace and a time marked by tolerance by the Sultanate of its Christian neighbors and subjects which is why any comparison between the Muslims of the Sultanate and the Aldmeri Dominion are completely inappropriate while comparisons between the Crusaders and the Thalmor are far more on point.
You're entitled to your own misinformed opinions but you're not entitled to your own facts.