No, The United States would be unrecognizable. It would slowly but surely turn into something similar to what the Soviet Union looked like before Ronald Regan was president.
I'm not exaggerating about how bad the United States would be if there was no constitution. I'm not an expert on this topic but if we got rid of the constitution or if there never was one, the United States of America would not exist or function the way it does today.
Barack Obama has talked about wanting to change or upgrade the constitution because he feels it is outdated and flawed, when the fact is that we still have the most powerful and effective constitution in the history of the world. In simple english, he wants to change it because the constitution we have doesn't let him do the things he wants to.
I don't want to be ranting on forever, but if you want to understand your question better, I suggest actually taking the time to carefully and thoroughly read the constitution of the United States of America, you can find free transcripts of it online. Just Google USA constitution.