I'm constantly surprised by how often people on both sides of the partisan divide in American politics accept the frame that Democratic Party politicians want to make the United States more "like Europe." Ed Glaeser today gives us a version of the Republican argument that Europe is horrible and therefore you should vote Republican and I've already seen Democrats grumbling that he's selling Europe short. I actually do think he's selling Europe short in some ways (conservative economists' tendency to love market prices except when making cross-country income comparisons is ... interesting) but I more seriously reject both sides' tendency to see American partisan politics in this light.
Generally speaking, the Republican Party would like to bring a much more European-style approach to tax structure (lower rates on capital and corporate income, less progressive overall base) to immigration (less of it and more narrowly targeted on alleged skill needs) to macroeconomic stabilization policy (less Keynesianism, more real business cycle) and to church-state relations. What's more, if you compare the United States of America to the European Union as a whole things look even more GOP friendly. The EU has more income inequality than the United States and the German vision of a European Union that constitutes a single monetary and trade bloc with a common defense policy while devolving social welfare policy to its constituent regions represents state of the art Republican Party thinking about such matters.
That's all just to say that it's a bit complicated. American domestic politics are not the same as German or Greek or French domestic politics, and EU-level politics are a different thing altogether. And of course politics exists in each of these European countries.