I would say partially no and partially yes. It depends on if you are looking within a society or beyond a society. Social Darwinism to me refers to the unmitigated competition of lasseiz-faire capitalism - think Herbert Spencer and late 19th century America. Fascists and Nazis were anti-capitalist as well as anti-Marxist, in that their core theoretical concept was to end the class struggle and reassert the organic power of the nation-state based on racial and national solidarity. Despite smashing independent labor unions, social democratic parties, and Communist parties, fascist/Nazi parties were at their core a combination of working-class parties and paramilitary organizations. They had corporatist economic policies which restrained and directed the business community for the national interest and they certainly had large welfare states. So, I think the picture painted here, and I hope it's accurate, shows a rejection of Social Darwinism. It's focused on constructing a national unity which empowers the state and envisions the nation-state as an integral and organic unit.

However, it certainly seems the case that the view of fascists/Nazis with regard to competition between races/nations reflects a view of Social Darwinism. Philosophically, they saw the races/nations in the world as in a state of constant war, and they glorified war. It seems to be an explicitly race-based philosophy as well - the stronger races would dominate the weaker ones. But in that sense, it's not much of a departure from the European imperialism of the time, except that they had the audacity to apply the concept within Europe. And finally, as a counter-argument to the first argument I made, the national solidarity of fascism/Nazism was certainly limited in nature. There was indeed a "survival of the fittest" factor, especially for the Nazis, but it was more in the sense of an attempt to impose what they saw as racial purity to "improve" the nation, rather than a natural and free-wheeling competition (eugenics programs, targeting the physically and mentally disabled, targeting homosexuals, Jews, Roma, Slavs, etc.) And of course, when you combine their internal concept of racial purity with their external concept of racial war and conquest, you behold the sheer horror which was Nazism.