Can't be true since 1789. The US Constitution, Article VI:
Clause 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.
Now, states like TX, MD, PA, NC, and so on, require that an office holder believe in God or a Supreme Being.
Until approximately 1793, the Delaware Constitution required the following statement to be signed by all public office holders:
"I_________, do profess faith in God the Father, and in Jesus Christ His only Son, and in the Holy Spirit, one God, Blessed forevermore; And I do acknowledge the holy scripture of the old and the new testament to be given by Divine inspiration."
Also...John Jay, Noah Webster, James Madison, Sam Adams and etc. all made strong statements in regard to the requirement for public office holders being Christians.
"No man shall hold office who eschews pasta" placemarker