Rituals and customs

I personally don't see what the problem is with using rituals and customs from another culture. The article chosen to illustrate the question is a poor example, in my opinion. This isn't so much a case of somebody disrespecting another culture so much as an example of gross medical malpractice. Frankly, who cares whether the doctor was treating the custom of the sweat lodge with the proper respect when the real issue is that he seems to have absolutely no idea how to conduct himself as a medical professional. By all means go to a sweat lodge if you're not a Native American. I'm sure it's quite physically and mentally cleansing. Just be sure that your host has, oh, SOME idea what he's doing.
In general, I think it's perfectly fine to participate in rituals from another culture. By that account, people shouldn't be able to learn Asian martial arts if they're not of Asian descent. Nor should non-Jewish people attend a seder, even if they are invited to one, as my family was a few years ago. It's stupid. Learning about other cultures is a great way to become more worldly, and participating in holidays and rituals is a good way to do that. Just be sure that, if it's an event/ritual of some religious or otherwise deep significance, that you behave in a respectful manner. And that you don' go to a sweat lodge hosted by a quack doctor.