I think this goes a little too far. While the madrasas were indeed religious in nature, they did a fine job of teaching one Leonardo Bonacci the thereto obscure subject of al-jabr, and the professors granted their students the ijazah. And we would be remiss to ignore the proximity of the Pandidakterion ("all-teachings") in Constantinople:

https://en.wikipedia.org/wiki/University_of_Constantinople

Indeed, we can trace the following sequence of events:

1054: Great Schism divides the Byzantine Empire and Western Europe

1048-73: Holy Roman Emperor Henry III attempts to dominate the election of popes

1050-80: Bishop, later (1073) Pope Gregory VII initiates the Gregorian Reforms, reigniting interest in the legal codes of Justinian, which gave legitimacy to the Church's autonomy contra the Emperor

1088: Irnerius begins teaching Roman law at Bologna, supposedly teaching the Four Doctors of Bologna (disputed by some historians), who would complete the medieval exegesis of Roman law in Europe

The principal innovation of the European university, which occurred sometime in the early 12th century, was the granting of degrees by the doctoral committee, rather than the advisor. As with most achievements in history, the development of the university was a gradual process, not a single event. The incorporation of students and professors into the universitates scholarum and colegia doctorum was initially necessary as collective bargaining against the mercurial and often conflicting state and church authority in medieval Europe, but eventually proved to maintain a higher standard of quality than had been previously possible.