There is really only one right to exist when it comes to a nation. It comes from the barrel of a gun. America's right to exist as a nation came at Yorktown.
The first sentence of the D of I says that America's right to Independence comes from the "laws of Nature and of Nature's God". Our country's founders were not criminal. This idea that America's right to exist comes from the "barrel of a gun" would make them criminal, because they were stealing the King's property.
America's way of life is dependent upon the "rule of law", not upon the "rule of men". This demands a respect for the written word, and that's why we're a nation full of lawyers and judges. It's a system which has allowed our country to progress and become the world's superpower. But in order for this system to be possible, we first must be able to believe in the legality of our country's existence. Because if our country's existence was illegal, the lawful thing for our government to do would be to hand America back over to its rightful owners.
The D of I broke the laws of the British Empire, which had soveriegnty over America at this time. In order for the D of I and America's independence to be legal, the "laws of Nature and of Nature's God" must have legitimate legal authority. The question at this point becomes "what God is the D of I referring to?"
In various different acts, the Continental Congress stated a belief in the Christian God. The Treaty of Paris positively identifies the God the US government believes in as being the "undivided Trinity", which is the God of mainstream Christianity. The law book of this God is the Holy Bible.
Genesis 11:1-8 tells us why God has divided the Earth into nations. This short passage from the Bible is the authority from which the USA derives its legal right to exist. Americans today can point at this law and claim that when God sent Englishmen to America, He was trying to build a new nation, not a world empire for King George.
If America's government has lost its respect for the Holy Bible's authority, this law is no longer good for us, and the lawful thing for it to do would be to turn the land back over to its lawful owners. It could either do that, or become a lawless society ruled by men.