Not speaking based on what I've grown up but this seems slightly more common on the American left than the American right. That said, examples of squabbles of similar forms on the right include over religion such as arguing over whether voting for Mitt Romney was ok given that he was a Mormon. (See e.g. here, with similar attacks on Glenn Beck. Recently one had certain aspects of the Tea Party call for a boycott of Fox News for being too pro-Obama. Similarly, of the Protestants on the right are still not ok with Catholics although they aren't a very large group and seem to be getting smaller. There's also a running trend in the fight between the more interventionist end of the right and the more isolationist end. See e.g. here. Another example is when Rick Perry tried to get HPV mandatory vaccination in Texas, there was blowback from the right as well as from the general libertarians.

But it seems that overall, these sorts of fights occur at a smaller scale than they do on the left. They don't involve as much splintering of organizations. And like many of the similar issues on the left, few people who aren't personally involved are paying much attention to them and even when one does, the differences often look small to outsiders even as the arguments get very heated.