I'll put it very simply: would people talk so much about sexism against women if females were the physically stronger gender? When women complain about not being taken as seriously as men, it's clearly because men are biologically stronger on average, yet how often would people accept that as an excuse? Would any politically correct person even try to bring this up as a valid argument?
Fact of the matter is that there are racists, sexists, etc. all over the planet. This is because prejudice in general is a human trait.
As you'll see here, involving biology in talks of discrimination isn't a new idea:
The only difference is that society is offended by this notion, so they blame it all on things like religion and cultural upbringing. They'll start a thousand counter-studies and countless debates to prove how it's 100% society's fault. On the other hand, most debates and studies about stature only serve to defend and excuse height discrimination.
Despite the derogatory stereotypes about short men (Napoleon Complex/Short Man Syndrome), negative media portrayals, and the language itself ("head and shoulders above, stand tall, man of his stature," etc.), rarely will you see anyone entertain the idea that heightism has cultural links. Here's the catch: go to a random third world place without western influence, and see if they know what Short Man Syndrome is. Meanwhile, in our society, if you stand against heightism, then you're the spoilsport, you're the bad guy for upsetting the status quo.
Hypothetically, if I said it sucks being in a racist society, people these days have been trained to wag their fingers at the racists. Replace race with height, and suddenly the problem is being short, not being heightist. Most people these days recognize society's impact on prejudice, not because they figured that out on their own, but because they've been taught as much. If they were actually clever enough to figure all this out, they would realize height is not so different. Relative Privation also comes into play here, but that's a whole different topic.
Point is, society and what it teaches has changed, not human nature. I guarantee if most of these "prejudice is learned" people had been born ages ago, they would be the bigots they despise. Rather than refraining from discriminating out of the goodness of their hearts, they clearly only do so when society tells them to. Why else would so many people (who don't consider themselves bigots) tear into short men without empathy? Why else would topics like fat shaming or big tits in video games have more people outraged than heightism? Society tells the populace what to be offended about, plain and simple.
At the end of the day, it doesn't even matter where bigotry comes from, as it should have no place in society. If a study came out showing how people view a certain race as uglier on average, it wouldn't excuse discrimination against that race. The urge to start a brawl is instinctive yet we're told to control it. Teenage pregnancy is natural but is it not a problem? Of course, it's "just different" with height.
The few studies about height and societal influence.