Aaron has oversimplified the Semmelweis story in some material ways:
* Semmelweis didn't institute "handwashing" in Vienna hospitals. Contrary to the conventional wisdom, which suggests that doctors in the 1840s were sticking horse-manure-covered hands into the exposed wounds of patients, handwashing was apparently already a norm. What Semmelweis did differently was to use lime to wash hands.
* Semmelweis' actual theory of the cause of childbed fever was wrong, and it was wrong in ways that made his recommendations hard to take seriously. Semmelweis' contention was that "cadaveric particles" were making their way into patients, and that those particles could only be removed by lime. But doctors observed cases in which no contact with either cadavers or injected or symptomatic patients lead to the same cluster of illnesses. It was thus difficult for Semmelweis to make a "scientific" case for why the lime worked; it obviously didn't help that he was wrong about why it did (his work predates the germ theory of disease, which would have taught him that rather than lime being effective at removing specific particles, it was instead effective at killing bacteria).
* Aaron's story (and Ayre's) has a heroic Semmelweis pleading for doctors to simply wash their hands in a specific way to save lives. But that's not necessarily what Semmelweis was arguing. Instead, the case he could have been making, loudly, was for an actual, specific, incorrect cause of childbed fever.
* Semmelweis himself was, apparently long before he lost his post, a notorious asshole. It did not help his cause that instead of carefully reasoning about the actual evidence, he instead seized on a single explanatory theory of childbed fever and then demanded (often by barging into hospital wards and berating the staff) that his peers adhere to it.
The point is not that Semmelweis didn't make an important discovery, or that we shouldn't be mindful of warped-sounding new knowledge that contradicts our existing theories. Of course we should be objective when considering facts that threaten our existing theories. But there's a reason John Snow and Joseph Lister [and Pasteur] are better known in the development of the germ theory of disease, and there's more to learn from the Semmelweis story than how the audience to a new theory should behave.