I wonder if, for example, the Catholic Church would marry two people under Canon Law without a state-issued marriage license. I mean, it seems to me that the secular institution stands in so many ways athwart the Catholic understanding of marriage that one would think that the Church would prefer to avoid any association with that institution in the public mind. I think that other Christian denominations might feel this way. If memory serves, the great R.J. Rushdoony (my favorite Protestant theologian) had something to say about this.
I’m not Catholic, but I checked into getting married without a marriage license. Apparently it is illegal for a priest/minister/etc. to do so.
They might in the future. They may only agree to marry people who wouldn’t get married by the state.