No, two witnesses are required because it's a serious contract, and serious contracts generally require witnesses. In this case, one reason for the witnesses is to avoid fraud - such as, say, a man giving a fake name to get a woman into the sack, as used to be not terribly uncommon.

There is no "spiritual" requirement for civil marriage; any such requirement would be blatantly unconstitutional.

Keep your religion at home folks, or in your church. It doesn't belong in our courthouse.