It is true that Aristotelian texts made their way back to Europe by way of the Islamic world. Indeed, Averroes's and Avicenna's commentaries figure into the works of Thomas Aquinas. For example, he uses Avicenna's argument for the distinction between essence and existence to perfect the understanding of God as Ipsum Esse Subsistens, or self-subsisting being whose very essence is "to be".

However, I wouldn't overstate the influence of Islamic civilization. Whereas the rationality of God and the created order, and thus its intelligibility, is central to Christianity (see John 1:1, where God in the second person of the Trinity is identified as the Logos), Islam is rooted in a voluntarist notion of God. The latter is quite hostile to any bold and sustained venture to grasp reality. Indeed, you don't see a sustained scientific enterprise anywhere in history save in the Christian West, and as Stanley Jaki among others argue, this is not an accident. Yes, you have sparks and flashes here and there across all sorts of civilizations, and yes, human beings by nature have a desire to know things, but as a sustained and confident enterprise, you need powerful convictions, like

1) the world is thoroughly and foundationally intelligible by virtue of its rational first cause,

b) human beings are capable of grasping this intelligible world as rational agents,

c) it is valuable to pursue such knowledge,

d) we are "permitted" and morally justified in such pursuits, and

e) progress in knowledge has a purpose.

Most other civilizations lacked some ingredient that led to either lack of desire or some kind of timidity about the whole thing: capricious gods, a chaotic and absurd universe, the world as an endless cycle going nowhere, pantheistic notions that make examining the world seem categorically offensive and sacrilegious, hedonistic cultures hostile to reason, etc. Even the Greek enterprise basically fizzled after Aristotle. In the case of Islam, I am reminded of a description that is relevant: Allah is an exalted caliph who, coming upon a fork, does not know whether he will go left or right. Yes, we do find great Islamic scholars, but the development of Islam has not produced conditions favorable to such sustained pursuits.

Where Scholasticism is concerned, it is not a singular view, of course, but a kind of enterprise with common features and shared foundational principles and one that also laid the intellectual groundwork for science, and sure, Averroes was studied, but that's also part of the point: the Christian tradition, best exemplified by the Catholic tradition, is omnivorous, interested in truth wherever it is found. The notion of logos spermatikos, or "seminal reason", is appropriate here. Truth is truth, wherever it comes from. To quote Philippians 4:8: "[W]hatever is true, whatever is honorable, whatever is just, whatever is pure, whatever is lovely, whatever is commendable, if there is any excellence, if there is anything worthy of praise, think about these things."

I will also add that it is untrue that "the Church" was in the business of "harassing scientists" or that the Inquisition was in the business of "killing the practitioners of science". Even the use of "scientist" or "science" in this was is anachronistic. You gave the stale example of Galileo as an example. This had nothing to do with science. Galileo was a man known for picking pointless fights, an irascible temperament, and a tendency to insult, and what we call the "Galileo affair" was not some oft-romanticized singular event caused by a clash between the Noble Galileo and the Mean Old Church, between the forces of Science and Enlightenment on the one hand and the forces of Darkness and Superstition on the other, but something that played out over decades of intermittent crankiness. It was, truth be told, incredibly boring, and one needs to go to great lengths of dishonesty to sex it up for IFLScience groupies. A better template for understanding what actually happened is the stereotype of bickering Italians with big egos than some cosmic battle between good and evil. In the case of men like the lecherous, arrogant, crank Giordano Bruno, he was not executed for "science", and he was hostile to the kinds of reasoned processes that science would employ, so it is absurd to canonize people like as some kind of "martyrs for science".