No, they aren't "Christian based beliefs" though. Just because someone is Christian, does not make everything they believe "Christian" in nature. Other people from other religious beliefs have thought of and believe that many of the ideals within the Constitution are good things to live by, the same way that many cultures and religions have had a main belief that people should "treat others as they wish to be treated", even long before Jesus said this. That belief was out there already. Jesus didn't invent it. Which means Christianity did not invent it.
Show me where in the Bible exactly that it says that a man can worship any God he wishes? Pretty sure one of the Ten Commandments, the very first one (plus the one on idolatry) contradict our very right to freedom of religion. Taking the lord's name in vain violates the right of freedom of speech. A person has a right to disobey and even dishonor their parents, for good reasons or bad, without punishment for that. You have a right to "covet" your neighbors stuff. In fact, this is a major factor in the success of capitalism. "Keeping up with the Jones'." As I've said, laws against killing of some kind (murder) have existed long before even Judaism, and killing someone violates their right to life, if it is considered "murder". Stealing violates a person's right to property. Lying is allowed, since it would violate our right to freedom of speech if it wasn't, up to it being able to cause significant harm to another person, such as in court or lying in some official capacity. Adultery laws which are used to criminalize sex outside of marital vows could likely never hold up in court now after the Lawrence decision. There are no fire arms in the Bible, and I doubt ownership of weaponry was ever really an issue covered directly by the Bible. Due process involves fairness at trials, and while likely mentioned in the Bible, was not actually practiced in Christian religions in general. It is a concept created by men, not the Bible or Christianity.
You are attempting to give a religion credit for the thoughts of men, for the ideas of men. That is wrong. The religion itself is not responsible for the ideas those men had. Plus, many of our founding fathers were deists, meaning that they didn't necessarily believe in everything the Bible or specific Christian denominations put out, said, believed. This, along with the fact that many more people have contributed to our ideals and rights in this country, which are not based on purely Christian beliefs, shows that our ideals do not come from Christianity. In fact, the South argued for a long time that they were completely morally fine with God in accordance with Biblical teachings in keeping slaves because such a practice is allowed in the Bible.
Oh, and you brought up the Ten Commandments, which are laws/rules, not rights. And, as I've pointed out, they are laws/rules in which many (at least half) violate guaranteed rights in the US Constitution.