shoot, I was wrong.
Yes, a president can technically run America without placing their hand on the Bible during the oath of office. While it is traditional for the president to do so, the U.S. Constitution does not require the use of a Bible or any religious text for the presidential oath of office. The Constitution only specifies the wording of the oath and the requirement for the president to take it before assuming office. The oath, as stated in Article II, Section 1, reads:
"I do solemnly swear (or affirm) that I will faithfully execute the office of President of the United States, and will to the best of my ability, preserve, protect and defend the Constitution of the United States."
If a president chooses to affirm rather than swear the oath (which is an option for those who prefer not to take an oath on a religious text), they can do so. The use of the Bible or other religious symbols is a matter of personal preference or tradition, not a constitutional requirement.