The public has a right to be accomodated by those operating a public conveyence. This is America and we set the rules. If someone has religious or other objections to the commonly accepted standards of a secular American society, then they need to find another line of work.
I am appalled that this issue is getting any more than a Donald Trump level of consideration: You're Fired.
Yup, the other day I got into a cab driven by an Amish guy. It was horse-drawn. When he saw I had zipper on my trousers he refused the fare. Boy, was I steamed.
I agree with you, if they have religious objections to picking up their fellow Americans, they should get out of the cab business. Simple as that.