Well, the Byzantines really deserve most of the credit for preserving the legacy of Greco-Roman science, arts, and letters. While it's true the Muslims did make their own significant contributions, they were rather selective in their stewardship, often focusing on practical sciences such as medicine.
Yes and who conquered the Byzantines?
Think narrated prologue from a 50's/60's CinemaScope epic! As for the influence of Constantinople, it's hard to overstate in my opinion; certainly when we're talking about western Christendom during this time-frame, there was nothing to rival it. Constantinople was rightly regarded as the "Queen of Cities." If we're examining the opinions of the Islamic world, we must acknowledge that while they now looked to their own cities such as Baghdad, Mecca, Damascus, etc., Constantinople was still greatly admired and recognized as a preeminent center of trade and culture.