Note that Social Darwinism is usually actively disavowed by biologists, including Darwin himself. Actual Darwinian natural selection states that organisms possessing properties that are better suited for survival in the environment tend to pass on their genes more often than ones that are less fit for the environment. Take for example a population hit by a disease. If 90% of that population dies due to that disease while the remaining organisms were resistant against it, the remaining organisms will reproduce and the race as a whole will be resilient against the disease in the future. This is something seen all the time in bacteria and insects, which reproduce in greater quantities and more quickly than the more complex mammals. In addition, random mutations occur and will either spread throughout the population or die out. This usually occurs slowly over the course of several generations. It is not necessarily the strongest/most ruthless/etc. who are the fittest, but it can be (and often is) that which can band together in groups for mutual benefit which turn out to be the fittest. It's easier to remember if you consider the context is "survival of those most likely/able to have children."
"The survival of the fittest" is also something of a Beam Me Up, Scotty!, as it was coined not by Darwin himself but Herbert Spencer (though Darwin added the phrase to the fifth edition of On the Origin of Species). Spencer actually did hold many views now associated with Social Darwinism such as that social programs should be dismantled as they would help the "unfit" to survive, but he didn't advocate eugenics and also strongly opposed imperialism, two positions usually put in the same camp. Many have noted that it should have been termed "Social Spencerism" if anything but Darwin was far more well known thus his name got stuck to it, and Spencer's actual philosophy was more complicated than simple Social Darwinism, making even this somewhat unfair. Blame can also be put on Francis Galton, half-cousin of Darwin and inventor of the Intelligence Quotient (IQ) Test, who coined the word "eugenics" and started that movement. Regardless, even if Darwin really had accepted these views, no one else would be obliged to, whether or not they felt his theory was true.
Darwin's "natural selection" was not the brute-force winner-take-all Lensman Arms Race contest some have characterized the process as. Natural selection isn't limited to a competition between different individuals or even species as to which is the most powerful. If predators became efficient enough to hunt all their prey to extinction, they would not be "fit", as they would die out too. All parts of entire ecosystems must fit together in a balance that could almost be likened to cooperation in social terms — the opposite of what the Social Darwinist believes. This also means that, to really emulate the principles of evolution, a conscious agent would have to weigh the consequences of their actions on everyone and everything else rather than fostering an individual or group that is capable of surviving at the expense of others.
Actual evolution can, in fact, lead to arms races that are self-destructive in the long term. You know saber-toothed tigers? Creatures like that have evolved multiple times, in response to their rhino-like prey developing thicker hides to protect against the longer teeth. In the end, the short-term advantage of keeping up with your main predator/prey would be countered by the disadvantage of lugging around such big teeth and hides and both competing variants would die out. Further, though evolution has found ways around it, the same problem applies with cheating versus co-operation: everyone will be better off if a group works together, but a cheater will always be individually better off than those who contribute, at least until the system collapses from too many cheaters (the way around this is by punishing cheaters, making it not be the more profitable option). The Social Darwinist is rarely found taking this into account. If they took destructive arms races into account, they would have to either knowingly do what's right by other standards than "natural selection" or knowingly act in a way that leads to an "evolutionary" dead end.
Social Darwinism in the real world has also been criticized for being tautological because what is called fitness is nothing more than whatever the speaker likes. This also makes it hypocritical: you want to intervene with things only to advance that which you want to prevail while saying that interference to advance anything else goes against what is natural.
In fact, sociologists use the term "social Darwinism" not to refer to the improvement of one entire society over another via "survival of the fittest" but rather that aspects within a society that are popular or useful will continue to exist, whereas the aspects which are unpopular or useless will be phased out. This is sometimes known as Memetic Mutation, as Richard Dawkins originally coined it: memes that are more influential/useful/likable/popular/easily imitated can and will spread from person to person, while memes that are old/forced/maladaptive get phased out. Quite ironically, social Darwinism is responsible for the growth of human and individual rights — exactly the opposite of what fiction implies it is for.
Additionally, since actual Darwinian natural selection can adapt only to your current situation, the greedy "only the most physically fit survive" version embodied by the villains of this trope is actually a poor long-term survival strategy — you would end up with a large number of almost-identical, seemingly-optimal "perfect" specimens that are then promptly slaughtered en masse once something else evolves (or some other change to their environment occurs) that preys on one of their shared weaknesses (or just kill each other off in the endless rounds of infighting they think is so important). An example of this in action would be deer populations. For millennia, evolutionary pressure made it optimal that bucks have large antlers, which allowed them to defend themselves and challenge other deer in fights over mates. Come the rise of sport hunting in the last couple of centuries, however, and all of a sudden deer with large antlers are being targeted for slaughter. Consequently, the average size of deer antlers plunged, and fast. Same goes to elephants who are now evolving to have no tusks, as the formerly rare tuskless elephants are increasingly selected for as the others are killed off.
It works out well then that evolution appears to be in many ways slow and conservative, frequently retaining seemingly "non-optimal" or currently useless genes through recessive traits and other mechanisms that can become useful in the future; for a species, diversity is more valuable than individual strength, something that racist Social Darwinists obsessed with "racial purity" have no understanding about (such a "pure" race like they propose would be genetically quite vulnerable, as has been seen with purebred dogs and inbred human royal families who suffered from genetic disorders).
In fact, Social Darwinism predated the Origin of Species by nearly a decade and for decades its proponents actually rejected Darwin's theory until they figured out using the name of scientists and justifying your actions with "For Science!" provided unquestionable PR.
An obvious, if unethical, way to disprove the fundamentals of Social Darwinism would be to kill the Social Darwinist. It's not possible for a less fit specimen to do; if it can be done then it proves you're more fit to survive than they are. But it should be easily understood the power of guns or cooperation or money or any number of things including just wanting it more than the other guy that can enable one person to so conclusively demonstrate their superior ability to survive compared to the other person does not make them a more evolved being, more valid or actually better in any way other than being better able to murder the other person. Nnote: this is a thought experiment. Don't kill people.
A final note: despite the similarity in spelling, this trope has nothing to do with Socialism. Part of the problem comes from conflating socialism with atheism (the two do not always go together), and then reasoning that someone who does not believe in God will thus take a crudely biological view of humanity, seeing them as animals with an instinct to dominate one another. But the fact is that the socialist's obsession with Equality above all else rules out - at least in theory - the prospect of him or her becoming concerned with genetic superiority.
Most Social Darwinists have naturally been of the Conservative or Libertarian orientation, while communists such as the Soviet Union condemned the Darwin-Mendel model of genetics in favor of a primitive Lamarckism, due to the latter being more attuned with Marxist ideology (this had disastrous consequences in agriculture). One must also not confuse Social Darwinism with eugenics. While many Social Darwinists did advocate eugenics (the Nazis most notoriously), it was also advocated by prominent progressives and socialists in the late 1800s/early 1900s, arguing that the "genetically diseased" should be relieved of having children they couldn't care for and that would be a burden to society, and that if a child is too disordered to even survive in the world then it is much better to subject them to euthanasia for the greater benefit of the whole; in other words, a cruel form of utilitarianism.
Another thing Social Darwinists failed to take into account is the concept of civilization. Though it's true that despots were often the first type of rulers, a civilization typically survived and flourished when every person, ranging from farmers and laborers to officials and rulers, worked together. Wounded individuals were cared for to restore them back to a functional capacity and rulers protected their citizens from raiders in exchange for loyalty and taxes. In diplomacy, though civilizations often viewed themselves as superior, even they saw the value of cooperation and peaceful coexistence, which led to things such as commerce, cultural exchange, and alliances. In other words, Social Darwinism would not be compatible with human dominance and survival. Everyone was born weak, for one, plus many people in history have been physically feeble, had few children (or none at all) yet also contributed highly to civilization (ironically, this is often the case among the most intelligent people, whom Social Darwinists frequently lionize, though by real evolutionary standards they would be unfit-not that this has to dictate attitudes).