The movie brings up, and does not resolve, a primary issue when it comes to the use of power (or in this case, superpowers): who should decide when, or if, it is used. With a person whose motives are impeachable, such as Captain America, it makes sense that they should be able to go and help when and where they are needed. Yet what about motives that aren't quite as pure? What about the collateral damage? Even if a committee is put in charge, how might their motives and agendas affect the situation? What if more suffering is caused due to the inaction of the people who could have done something about it (e.g. Spider-Man's origin dilemma)? The problem is that both sides of the argument were right. How they handled the situation might be in question, but the motives and ethics behind both sides ring true.
I suppose these were the questions that the founders of our Constitution struggled with as well. While the Constitution is not a perfect document, I like to think that they did a fairly good job at trying to come up with good answers. Our Bill of Rights, in particular, is a testament to that.