Honestly, yes it does. As much as I'd like to say it's just a piece of fabric, to me it stands for something more than that. When people of other countries do it, I can understand that they're trying to protest something, and whether I agree with them or not I can respect that. It's more when the people in my country (I'm American, by the way) show the flag disrespect; dragging it on the ground or whatever else.
You can say it's just a piece of fabric, and you can say that the fact you happened to be born on a random piece of land within the States doesn't matter, but to me it does. I believe in what the country stands for; not what people will construe from what the government has done in the past, or what the stereotype for what Americans is, but what the country was founded on, what it's supposed to stand for on freedom, acceptance, and equality.
The flag has been so important in the past because it served as a symbol of unification. Obviously things are not perfect, and there have been mistakes made, but the flag itself, in its base form, serves to show that the people of the United States, be they originally from Europe, Asia, Africa, or wherever else, all form a part of the single nation that is (on paper) devoted to making sure those people have an equal and strong opportunity to become the best they can be. Without the flag serving as that symbol of unification, what would it be? We all come from different places, we all live in different places, we have different accents, we have different cultures, and so on and so forth. The flag serves to show that, whatever combination of those things makes up you, you're a part of something bigger, along with your neighbors and their neighbors and their neighbors and so on, and you're not alone.
Rather idealistic of me, isn't it?