The Treaty of Tripoli disagrees, and I trust it as a rubric more than modern interpretation.
Yale law school, The Avalon Project, had a whole study on the Treaty of Tripoli. Article 11 is the offending addition. It was only in one copy of the treaty. I just checked their website and all that info, reams of it, have been removed. Surprised? I'm not.
As even a casual examination of the annotated translation of 1930 shows, the Barlow translation is at best a poor attempt at a paraphrase or summary of the sense of the Arabic; and even as such its defects throughout are obvious and glaring. Most extraordinary (and wholly unexplained) is the fact that Article 11 of the Barlow translation, with its famous phrase, "the government of the United States of America is not in any sense founded on the Christian Religion," does not exist at all. There is no Article 11. The Arabic text which is between Articles 10 and 12 is in form a letter, crude and flamboyant and withal quite unimportant, from the Dey of Algiers to the Pasha of Tripoli. How that script came to be written and to be regarded, as in the Barlow translation, as Article 11 of the treaty as there written, is a mystery and seemingly must remain so. Nothing in the diplomatic correspondence of the time throws any light whatever on the point