Now the real proof is in the Treaty of Tripoli. You can read the entire Treaty here, it is not long. The main point of interest is the first part of Article 11.
"As the Government of the United States of America is not, in any sense, founded on the Christian religion;"
Now that was written because the people of Tripoli were Muslims or Mussulmen, as they were called then. So announcing that the US was secular meant no enmity would be held against the Muslims from the US for their religion.
This article was read in its entirety to Congress. It was only the third unanimous vote recorded in US history. It was also published in the Philadelphia newspaper and two New York City papers. There was no public outcry. Founding father and then President John Adams signed the document.
The United States of America is and has always been a secular country founded on secular ideas and English common law.