The Constitution doesn't just "make no requirement of swearing in on any object," it explicitly prohibits what this dude is demanding.

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.

—Article VI