To quote the treaty of Tripoli

As the Government of the United States of America is not, in any sense, founded on the Christian religion,—as it has in itself no character of enmity against the laws, religion, or tranquility, of Mussulmen, and as the said States never entered into any war or act of hostility against any Mahometan nation, it is declared by the parties that no pretext arising from religious opinions shall ever produce an interruption of the harmony existing between the two countries.

You know, the founding fathers passed that. But it was John Adams, the average American's unknown president. They could probably tell you more about Hamilton, and he wasn't even president.

Our constitution does not mention God or Christianity at all, and implements several structural safeguards into our system of government to prevent the mixing of the secular government and religious life.

Off the top of my head, there is:

The No Religious Test Clause, Article VI, paragraph 3:

The Senators and Representatives before mentioned, and the Members of the several State Legislatures, and all executive and judicial Officers, both of the United States and of the several States, shall be bound by Oath or Affirmation, to support this Constitution; but no religious test shall ever be required as a qualification to any office or public trust under the United States.

This bars the implementation of any religious requirements to hold public office. Any law that states one must hold or practice any religious beliefs is unconstitutional per this clause, and is therefore unenforceable.

The Establishment and Free Exercise Clauses of the First Amendment:

Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.

In essence, Congress (or any state or local legislatures, after the 14th Amendment incorporated the First down to the states) cannot establish an official state religion, or endorse or otherwise promote a religion, religious belief or religious practice. Nor can it prohibit the free exercise of a religious belief so long as the practice does not infringe on the rights of others or otherwise violate the law. My civics teacher's go-to example for this was that If his religion involved running down the street naked throwing snakes at people, he would not be able to do so without risking arrest, since each of those is an act prohibited by the law, and the laws are not targeting his religion, but rather individual acts that represent a risk to others.

Also, the courts have implemented tests such as the Lemon Test named after the court case Lemon v. Kurtzman (1971) which outlines out how secular government is to interact with religious bodies in regards to legislation or other actions. The purpose of the test is to ensure that the government avoid unnecessary entanglements with religious entities, and itself remain a secular entity.