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What do you mean by a "Social Darwinist?" That's a fairly broad rubric and encompasses ideas that existed long before Darwin and movements which flourished after his death. Hobbes' Leviathan can easily be seen to have elements of Social Darwinism, two centuries before Darwin published On the Origin of the Species. Malthus, writing over a decade before Darwin was even born is clearly espousing ideas that fit with Social Darwinism. Even Swift's satirical exhortation for the Irish to sell their babies as food can be seen as reflecting ideas of the time that would also fall under the later title of Social Darwinism.

Similarly, Darwin's own contemporaries were espousing ideas on this subject independently of, and in reaction to, Darwin's works. Herbert Spencer, for instance, published Social Static in 1851, eight years before Darwin published his famous work. In it, Spencer wrote (on the subject of Poor Laws):

In demanding from a citizen contributions for the mitigation of distress—contributions not needed for the due administration of men’s rights—the state is, as we have seen, reversing its function, and diminishing that liberty to exercise the faculties which it was instituted to maintain... Of two individuals, one may use his liberty of action successfully—may achieve the gratifications he seeks after, or accumulate what is equivalent to many of them—property; whilst the other, having like privileges, may fail to do so. But with these results the state has no concern. All that lies within its commission is to see that each man is allowed to use such powers and opportunities as he possesses; and if it takes from him who has prospered to give to him who has not, it violates its duty towards the one to do more than its duty towards the other.

(Emphasis mine.)

Darwin's cousin and colleague, Francis Galton, in part reacting to On the Origin of the Species, wrote Hereditary Genius in 1869, wherein he stated:

The time may hereafter arrive, in far distant years, when the population of the earth shall be kept as strictly within the bounds of number and suitability of race, as the sheep on a well-ordered moor or the plants in an orchard-house; in the meantime, let us do what we can to encourage the multiplication of the races best fitted to invent and conform to a high and generous civilization, and not, out of a mistaken instinct of giving support to the weak, prevent the incoming of strong and hearty individuals. (p.357)

(Again, emphasis mine.)

Galton would not, however, coin the term "eugenics" until a year after Darwin's death in 1882, and what would become the Galton Institute was not founded until 1907. The academic journal of the Institute, the Eugenics Review would not publish its first issue until 1909. So even though Galton was espousing views and writings that could be called Social Darwinism during Darwin's lifetime, the most famous formulations of his ideas would not really come to fruition until after Darwin's death.

So the question remains: What did Darwin think about all these ideas happening around him, occurring at least in part in reaction to his theory of natural selection?

Darwin's most extensive writing on the subject occurs in his Descent of Man, first published in 1871. In particular, the section titled "Natural Selection as Affecting the Civilised Nations," which specifically references Galton's Hereditary Genius throughout. The opening paragraph of that section is often cited as proof of Darwin's complicity and support of "Social Darwinism." It goes:

With savages, the weak in body or mind are soon eliminated; and those that survive commonly exhibit a vigorous state of health. We civilised men, on the other hand, do our utmost to check the process of elimination; we build asylums for the imbecile, the maimed, and the sick; we institute poor-laws; and our medical men exert their utmost skill to save the life of every one to the last moment. There is reason to believe that vaccination has preserved thousands, who from a weak constitution would formerly have succumbed to small-pox. Thus the weak members of civilised societies propagate their kind. No one who has attended to the breeding of domestic animals will doubt that this must be highly injurious to the race of man. It is surprising how soon a want of care, or care wrongly directed, leads to the degeneration of a domestic race; but excepting in the case of man himself, hardly any one is so ignorant as to allow his worst animals to breed.

Most of the people citing this passage (it is popular in anti-evolution circles), however, leave off the very next paragraph, which starts off with this statement:

The aid which we feel impelled to give to the helpless is mainly an incidental result of the instinct of sympathy, which was originally acquired as part of the social instincts, but subsequently rendered, in the manner previously indicated, more tender and more widely diffused. Nor could we check our sympathy, even at the urging of hard reason, without deterioration in the noblest part of our nature. The surgeon may harden himself whilst performing an operation, for he knows that he is acting for the good of his patient; but if we were intentionally to neglect the weak and helpless, it could only be for a contingent benefit, with an overwhelming present evil.

Again, emphasis mine, but Darwin there is flatly stating that it would go against the most ingrained and admirable trait of humanity, our empathy, to treat our fellow humans as mere livestock. This is actually well in accordance with Darwin's writing on the evolution of humankind and the development of complex societies. He repeated stresses the importance of cooperation, sympathy, and altruism.

But wait, it gets more complicated, since Darwin concludes that paragraph thusly:

We must therefore bear the undoubtedly bad effects of the weak surviving and propagating their kind; but there appears to be at least one check in steady action, namely that the weaker and inferior members of society do not marry so freely as the sound; and this check might be indefinitely increased by the weak in body or mind refraining from marriage, though this is more to be hoped for than expected.

So, to bring this all together, Darwin accepts the idea that idea that there are the "weak" and the "vigorous" in society, but that neglecting the former would debase the core of humanity, and anyway, the weak don't produce as much regardless. Darwin, in fact, spends most of the rest of this section detailing how the "black sheep" (he literally uses this in an analogy) of society die earlier, marry later, and generally do not breed as much as the "better class of men." The degeneration of society posed by the poor, which led Spencer and Galton to militate against providing them aid and succor is, to Darwin, naturally kept in check.

Moreover, Darwin recognizes the limitation of biology in human society and the multifactorial nature of the rise and fall of societies. Take this passage:

It has been urged by several writers that as high intellectual powers are advantageous to a nation, the old Greeks, who stood some grades higher in intellect than any race that has ever existed (26. See the ingenious and original argument on this subject by Mr. Galton, 'Hereditary Genius,' pp. 340-342.), ought, if the power of natural selection were real, to have risen still higher in the scale, increased in number, and stocked the whole of Europe. Here we have the tacit assumption, so often made with respect to corporeal structures, that there is some innate tendency towards continued development in mind and body. But development of all kinds depends on many concurrent favourable circumstances. Natural selection acts only tentatively. Individuals and races may have acquired certain indisputable advantages, and yet have perished from failing in other characters. The Greeks may have retrograded from a want of coherence between the many small states, from the small size of their whole country, from the practice of slavery, or from extreme sensuality; for they did not succumb until "they were enervated and corrupt to the very core." (27. Mr. Greg, 'Fraser's Magazine,' Sept. 1868, p. 357.) The western nations of Europe, who now so immeasurably surpass their former savage progenitors, and stand at the summit of civilisation, owe little or none of their superiority to direct inheritance from the old Greeks, though they owe much to the written works of that wonderful people.