First off, until recently, there wasn't any strong pressure for it. Post War the American economy was booming, while the European economies were struggling. The Europeans needed to buy off the lower classes lest they suffer revolution, and they could afford to because they cut defense spending massively, and were subsidized by the Americans (the Marshall plan got their economies back together, and NATO let them let the Americans bear a major share of Europe's defense).
In addition, Americans tend to be far less trusting of their government then Europeans, in addition to being a far larger and less homogenous population then the European nation states. What works for tiny Denmark, does not extend to the varied American states. There are already programs in America for the poor for health and welfare, they just aren't universal. Senior citizens and the poor Have Medicare and Medicaid, so instead of covering everyone, the government only covers the most vulnerable sections of society (which ends up being rather cheaper). Most importantly, until recently, there was simply more upward mobility and prosperity in the American economy compared to the European economies, thus alleviating much of the pressure.
