Dawkins isn't a Social Darwinist, but it isn't at all clear how he justifies not being one. From Stand to Reason:
In a response to the segment in Ben Stein's Expelled that makes a connection between Darwinist thinking and the Holocaust, Richard Dawkins said the following:But then Dawkins "is smarter than you, he's got a science degree.":My own view, frequently expressed . . . is that there are two reasons why we need to take Darwinian natural selection seriously. Firstly, it is the most important element in the explanation for our own existence and that of all life. Secondly, natural selection is a good object lesson in how NOT to organize a society. As I have often said before, as a scientist I am a passionate Darwinian. But as a citizen and a human being, I want to construct a society which is about as un-Darwinian as we can make it. I approve of looking after the poor (very un-Darwinian). I approve of universal medical care (very un-Darwinian). It is one of the classic philosophical fallacies to derive an 'ought' from an 'is'.The problem for Dawkins is that if Darwinism - the foundation of his worldview - is true, then there is no "ought." An real ought can only exist in a place that has been designed for a purpose; only if there's a standard we need to meet - a standard outside of ourselves that we're obligated to - can there be an ought. In a naturalistic world, there is only "is," and yet even Dawkins recognizes that this is not the case with our world. Here's what he's saying: Natural selection is the explanation of all life and society ... and it is contrary to what we know to be true about life and society. In other words, there are things we know to be true about the world - about morality ("oughts") - that are contrary to what we would expect to find in a world without standards, purpose, or value.
Dawkins's sincere defense of "oughts" reveals the inadequacy of his own worldview. The best worldview is the one that makes sense of all of what we know to be true about the world - life, beauty, goodness, evil, morality, and yes, even science. Naturalism simply can't do this. ....
Stand to Reason Blog: Darwinism is an Inadequate Explanation, Vox Populi: Dick to the Dawk