You make a great point - that many of the discoveries were not directly or solely 'muslim' - but yet they were.
As you point out, many of of the key cities that fostered discovery were the locus of many ideologies, that were allowed to coexist. And that is the key point, at the height of the Muslim world the Muslim world was tolerant of local ideas, and learned from them, assimilated them, wove them in to their writings and scientific understanding, grew them in to something more.
It is dangerous to view any culture as an island unto itself. At least any culture that isn't isolated on an island. The truth is that Islamic cultures, like others, went through varying times of acceptance and rejection of other ideas. And like other cultures, the times of acceptance led to the times of greatest innovation, while the times of isolation and rejection led to war and stagnation. A number of Karen Armstrong's books lend great insight.
Makes you think a little about where the US is.