What to call someone who, apart from very inappropriate acts, doesn't care about what others think about him?
So, for instance, someone who doesn't occupy himself with the opinions of others and what they think of him, however who wouldn't go naked on the streets, because he completely doesn't care about what people think.
Solution 1:
I think we could call such a person Self-assured. You can also use intrinsically-motivated (meaning self-motivated in actions).
Solution 2:
Answers others have posted show that the word you use depends on whether you are thinking of this as a positive or negative characteristic. My first thought on reading your question was that the general answer would be that you call such a person "independent", or "independent-minded" if you need to distinguish from other forms of independence. Often we use longer phrases, like saying he "makes up his own mind" or "thinks for himself". I think those are all basically positive-sounding terms.
If his independence of thought leads him to actually do things that are seriously unconventional, than he may be a "non-conformist" as @hauron says. That word may be positive or negative depending on context.
If it leads him to being inconsiderate of the opinions of others, then he could be "insensitive" as @josh61 says. I don't think his other suggestion, "egoist", is quite right. Someone could disregard the opinions of others, not because he thinks he is better than them, but just because he wants to know the reasons why and examine things himself. Again, it depends on context and shades of meaning.
Solution 3:
Perhaps a free spirit or a nonconformist?
Solution 4:
One of the ways people differ is in their sensitivity to what others think of them. From a psychological trait standpoint, this is sometimes called self-reliance. Those who are highly concerned about the impression they make on others may be considered apprehensive, group oriented, afflilative, or joiners, whereas those who care much less about what others think of them are called solitary, resourceful, or self-sufficient. Those at the extreme end of self-sufficiency may even be considered antisocial.
For more discussion of these and other traits, see:
http://en.wikipedia.org/wiki/16PF_Questionnaire