I would argue that there were two major traits associated with the Progressive Era. The first of these traits was the belief that the middle class was superior to both the rich and the poor. The other trait was the belief that the government should use its power to impose middle class values on the rich and the poor. Both of these ideas are contrary to the ideas of Social Darwinism.
The Progressives believed that the rich and the poor both, in a sense, had something wrong with them. The poor behaved in ways that did not allow them to get ahead in society. The most important flaw was their propensity to drink. They also lacked knowledge about things that the settlement houses were built to teach them, things like proper nutrition and child care. They needed to be encouraged (or coerced) to behave more like the middle class. The problem with the rich was their greed and their willingness to harm society for their own benefit. The rich were willing and eager to build and own squalid tenements where the poor had to live. The rich owned factories where workers endured poor conditions and which, at times, turned out substandard and unsafe goods (think The Jungle by Upton Sinclair). The rich used their wealth to control the political process. They needed to be reined in so their greed could no longer hurt society.
All this challenged the ideas of Social Darwinism in two main ways. First, Social Darwinism stood for the idea that people should have to compete with one another for dominance. The government was not meant to enact laws to help protect the poor. Instead, the poor needed to compete and try to get ahead. Otherwise, they would just be encouraged to continue acting in the ways that made them poor, thus harming our society as a whole. Secondly, the tenets of Social Darwinism held that those who had gotten rich were the fittest in the society. They were the ones who had been more successful than anyone else. This meant that their actions were superior to those of less successful people. People who believed in Social Darwinism thought that the rich should be given more control over society since they were clearly the people who knew how to navigate society best. The Progressives challenged this idea by saying that the middle class was morally superior to the rich and that the middle class should be able to use the power of government to prevent the rich from doing as they liked. In these ways, Progressive ideas posed a serious challenge to the ideas of the Social Darwinists.