How is science unfalsifiable?
"Science" as a body of knowledge is definitely falsifiable. However, if you're using "science" in terms of the scientific method, using the word "unfalsifiable" makes no sense. "Unfalsifiable" is only something that can be applied to statements that could ostensibly be proven to be true or false, though in this case the statement is set up in such a way that you cannot ever prove it true or false.
Methods aren't statements, they're not something that can either be "true" or "false". You can't say something like "Logic is true" or "Mathematics is false". It makes literally no sense to talk about them in that fashion, since they aren't statements or observations that have truth values in of themselves. What you can do, though, is show whether a methodology is sound or not in terms of the statements it produces. Seen in that light, mathematics, logic, and science are very sound and useful indeed from the statements, predictions, and observations they have made.
Theology in of itself is not something that can be described as "unfalsifiable" either. The statements and claims it makes can be though.
Also, the argument that religion somehow be credited for the creation of such things such as the university, or the hospital, is an empty one to me. Most humans who have ever lived had religious convictions of one sort or another, by nature of how our psychology tends to have a propensity towards making such cognitive, in my opinion, mistakes.
I think it would be hard to prove that such institutions would've come about irregardless of which ever religion happened to be in vogue at a region or time. Did religions help shape the institutions as we know them in their early days? Undoubtedly. Were they directly responsible for them though? I don't think so. Taking care of the sick and elderly, and trying to educate people are hardly ideas or propensities that religion formed, instilled, or created. The inevitable institutionalization of these things would no doubt be paired with whatever else was conveniently institutionalized already.