We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.--That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed, -
Read Locke's 2nd Treatise. God is the foundation of natural law. Natural law is the foundation of the United States. Without God's natural law, law is arbitrary, man-made, and rights can be revoked at any time by any of man's governments.
Without natural law, rights mean absolutely nothing. Zero.
That's what "under God" means.