"Who invented the scientific method?" is not an easy historical question to answer because it's not clear a) that there is anything called "the scientific method" and b) that anyone specifically invented it. Let me elaborate a little bit.

The idea that there is a single investigative method behind scientific inquiry is totally false. That thing you are taught in high school (hypothesis leads to methodology leads to experiment leads to results, etc.) is just pedagogical claptrap meant to structure early education. It bears no real relationship to scientific practice either today or in the past. Pretty much all attempts to codify a specific, special method that explains how science "works" fail both historically and in the present day, because what we call "science" is a complex grouping of many different types of social activity in many different domains of knowledge. If you start to define it too specifically you find huge bodies of work (past and present) don't fit into it; if you start to define it too broadly you find that you definition includes things that you don't want it to (like astrology). This is what philosophers of science call "the demarcation problem" and it is considered basically unsolvable, especially by historians of science.

This does not mean that one cannot come up with a loose sense of what people mean by "science" in the 21st century: e.g., a social activity (in that it is done in networks and groups of people, even when the investigator is essentially solitary) that seeks to come up with naturalistic understandings of the world (e.g., as opposed to supernatural ones, though the natural/supernatural boundary is contestable) that are largely based on inductive and deductive reasonings (e.g., a "toggling" between theory and experiment and back again), typically (but not always) with a heavy emphasis on quantitative modes of analysis. What a mouthful! But that more or less describes what most people mean by "science," even if it's doesn't work in all cases and can apply to things that people think are not "science."

So, if we want to go with this sort of definition, who invented it? The answer is... no single people or culture or group. All of these pieces were assembled in different ways over literally thousands of years. The "authorship" includes many cultures, including those in the broader Mediterranean region, the Middle East, India, and even China. There wasn't one moment (much less one person) in which all of these elements were created. It's like saying that the idea of capitalism was "invented in the West" — I mean, some elements of capitalism sprung up there (e.g., the joint-stock company), but many elements were present elsewhere first and directly used by people in the West in their conception of what was being developed.

OK, so what's up with this "Scientific Revolution" thing you hear about in school? Long story short: the idea that Western Europe had this sort of unique moment of "inventing" science is mostly 18th- and 19th-propaganda (usually aimed at undermining religious institutions). Some things did happen in Western Europe, but there was no magical moment in which people became more scientific. Rather, what you see is an increasing size of a community of what we might call scientific investigators, an institutionalization of the means of research and circulation of knowledge, and the self-conscious adoption of "Science" as a category of human activity that was advocated as an alternative knowledge system from other categories (e.g. religion, politics, whatever). The institutionalization of this in the 18th- and 19th-centuries was coupled with stories about its uniqueness, special methodologies, difference from religion, etc., as well as its Western-centric nature.

One can make a more generous version of it by emphasizing the development of institutions of learning in Western Europe and the professionalization of scientific inquiry, and talking about how various norms changed over time. This is what careful historians of science do when they talk about the "Scientific Revolution" (often in quotation marks), referring not to some kind of disjunctive "invention" of science but rather to changing attitudes and norms over time. These people would be very hesitant to say that this was "invented" anywhere in particular, much less "the West."

The one tricky thing: sometimes those countering bigots (which Shapiro surely is, as well as a hack) can easily flip into a kind of "boosterism" themselves. "The scientific method" wasn't invented in the Middle East, either. One can talk about the important contributions of medieval Islamic scholars (esp. to the development of Western science) without somehow moving the magical moment of "invention" there.

Does all of this sound tricky and wishy washy and complicated? That's because it is — "science" is a tricky concept both philosophically and historically, because it's not a "thing" so much as a series of social institutions, philosophical mindsets, behaviors, norms, and so on. Anybody who desires to simplify all of this into some sort of simple story to back up a political position (good or bad, but especially bad) is committing a grave historical error, whether they know it or not.