This is a very common and accepted and required custom. Christians swear on the Jewish Bible and their additional writings.
Jews are allowed to just use the original Bible. Some people use family Bibles. Yes, even George Washington swore on a Bible over two hundred years ago.
Two hundred years may seem like a long time ago, but it's nothing compared to Jewish History.
Jewish History, yes, the history written in that Bible they all swear they'll live by.
The Story of Shiloh and the Jewish Nation are in the Bible, yes that book that smiling Michelle Obama is holding while her husband, Barack Hussein Obama places one hand on, that same Bible, and raises his other and promises, swears on the Bible to be a good President of the United States of America.
Now for a small question, a minor detail:
If he makes demands that deny the validity of the Bible, what does that do to the legitimacy of his presidency?
Just one of my dumb questions. Food for thought.