That's not easy to answer, because it is in the nature of American political parties that they cover a broader political spectrum than political parties in Europe usually do.
That being said, the GOP roughly covers the spectrum of AfD, CSU, and at least the CDU's conservative wing. However, I would say the CDU also partially overlaps with the Democratic Party's more centrist wing.
In addition, since the GOP has shown an increasing tendency to embrace more extreme positions in the past decade or so, I'm also inclined to say that the GOP's ideological center increasingly aligns where the AfD stands in Germany.
In short, both the Trumpists and the Freedom Caucus would found themselves at home in the AfD (as the AfD's economically populist and the AfD's more "libertarian" wing respectively). More moderate types like Kasich, McCain etc. would certainly belong into the CDU.
Who in the Democratic Party is as anti-SSM as most of the CDU?
Maybe that's the wrong question.
More than a quarter of the CDU voted in favour of legalizing same-sex marriage, including defense minister Ursula von der Leyen, head of the Chancellery Peter Altmaier, and CDU secretary-general Peter Tauber.
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