First, the Muslim world benefited greatly from the Greek sciences, which were translated for them by Christians and Jews. To their credit, Muslims did a better job of preserving Greek text than did the Europeans of the time, and this became the foundation for their own knowledge. (One large reason for this, however, was that access by Christians to this part of their world was cut off by the Muslim slave ships and coastal raids that dominated the Mediterranean during this period).
Secondly, many of the scientific advances credited to Islam were actually “borrowed” from other cultures conquered by the Muslims. The algebraic concept of “zero”, for example, is erroneously attributed to Islam, but it was, in fact, created by the Hindus and merely introduced to the West by Muslims - along with the products of other cultures that were found to be useful to their new rulers.
In fact, conquered populations contributed greatly to the history of “Muslim science” until gradually being decimated by conversion to Islam (under the pressures of dhimmitude). The Muslim concentration within a population is directly proportional to the decline of scientific achievement. It is no accident that the Muslim world has had little to show for itself in the last 600 years or so, since running out of new civilizations to cannibalize.
Third, even the great Muslim scientists and icons were often considered heretics in their time, sometimes for good reason. One of the greatest achievers to come out of the Muslim world was the Iranian scientist and philosopher, al-Razi. His impressive works are often held up today as “proof” of Muslim accomplishment. But what the apologists often leave out is that al-Razi was denounced as a blasphemer, since he followed his own religious beliefs – which were in obvious contradiction to traditional Islam.
Fourth, even the contributions that are attributed to Islam (often inaccurately) are not terribly dramatic. There is the invention of certain words, such as alchemy and elixir, but not much else that survives in modern technology that is of any practical significance. Neither is there any reason to believe that such discoveries would not have easily been made by the West following the cultural awakening triggered by the Reformation.
As an example of this, consider that Muslims claim credit for coffee, since the beans were discovered in Africa (at the time, an important source for Islamic slave trading) and first processed in the Middle East. While this is true, it is also true that the red dye used in many food products, from cranberry juice to candy, comes from the abdomen of a particular female beetle found in South America. It is extremely unlikely that the West would not have stumbled across coffee by now (although, to be fair, coffee probably expedited subsequent discoveries).
In fact, the litany of “Muslim” achievement often takes the form of rhapsody, in which the true origins of these discoveries are omitted - along with their comparative significance to Western achievement. Scientific, medical and technological accomplishments are not something over which Muslim apologists want to get into a pissing contest with the Christian world. Today’s Islamic innovators are known merely for turning Western technology, such as cell phones and airplanes, into instruments of mass murder.
To sum up, although the Islamic religion is not entirely hostile to science, neither should it be confused as a facilitator. The great achievements that are said to have come out of the Islamic world were made either by non-Muslims who happened to be under Islamic rule, or by heretics who usually had little interest in Islam. Scientific discovery tapers off dramatically as Islam asserts dominance, until it eventually peters out altogether.