The reason why Native Americans might be offended by the term New World is because they were living here for centuries before the Europeans came and colonized "New World." It wasn't new and so for the Europeans to call it the "New World" implied that the Native Americans are worthless and their existence doesn't count. The term "New World" suggests that the Europeans just take whatever they find and claim it as their own, completly disregarding the native peoples who were there before them.
