Before that, however, I don't really get the sense that "science" had any real connection to private property at all. For a period of about 300 years starting in the late 15th century, I think science was studied as a "pure" subject simply for the sake of satisfying someone's thirst for knowledge -- not for the sake of monetary gain. People like Galileo or Sir Isaac Newton had no patents or copyrights that allowed them to derive financial benefits from their discoveries. In fact, it was generally quite the opposite . . . science was such a "pure" subject matter and presented such poor prosoects for financial gain that the pursuit of scientific research required one to either secure the backing of wealthy benefactors for financial support, or work with a religious or academic institution whose primary function was the pursuit of abstract knowledge.
Perhaps there is a clear way to define exactly which forms of science required private property rights and certain government/civic systems to thrive. I'd say that advances in pure sciences required nothing more than the right minds and the proper foundation of knowledge, while advances in applied sciences usually required some tangible financial gain on the part of the person or institution doing the research.
What's most interesting about this second point is that our modern Western social/government systems have also provided the basis for a number of things that are gradually driving the collapse of science, including the use of blatantly incorrect applications of "science" to" 1) reap enormous financial gains and 2) promote political agendas.
For while applied science depends on security of private property, pure science, again, as you pointed out, relies on "idle" wealth -- either your own [the scientist] or someone else's [outside investor]. That, in turn, means that there must be capital over and above what is required for living. IOW, only a prosperous society can afford the "luxury" of pure science.... And that leaves the Islamic world o-u-t, out.
Certain people in the Islamic world have unimaginable wealth, but Islamic society,as a whole, is becalmed in the stagnate waters of poverty. And as wealth sleeps uneasily on a bed of poverty, money that (theoretically) might go into funding pure science is, instead, spent on bread and circuses.
In closing, to address the puzzlement over the concept of "private property" has been in the Western dominance of science and technology bare in mind that Applied Science, unlike her more ethereal sister Pure Science, is driven by industry and its money which, in turn, is made secure by legal protection of private property.