Although, a lot of conclusions are drawn from this fact, the reason is more based on American history than our own ideals.
In early American history, during the revolutionary war, the colonists did not have a standing army. The people who fought against the British for freedom were ordinary colonists. Farmers and tradesmen, not soldiers. These were called militias, organized groups of civilians. If you ask the NRA (American pro-gun association) they will probably tell you it's because the founding fathers wanted the contutionalize a method for future generations to defend themselves against the possibility of a corrupt government. This may or may not be true, but even after the colonies were originally freed, the United States was not a safe place. They were quite dangerous as many indian tribes still existed (Not all bad, some peaceful, some violent, and some reasonably aggressive due to past conflicts).
So the US was founded in such a way that all citizens needed the right to defend themselves. How do you remove such a right when guns are already so common? It's not easy, as the moment you do, the good citizens no longer have their arms, while the criminals will keep there own.