Ah, but England had (and in theory still has) an established Church.
That was also how it was in the Roman Empire from sometime in the 9th century when the Roman state decided that civil marriages were redundant, onward — before then, a Christian marriage was a Roman civil marriage undertaken with the blessing of the local bishop, followed by reception of the Holy Eucharist at the same Liturgy by the newly civily married couple. There was no Christian marriage ceremony. (Yes, there was still a Roman Empire then, its capital was at Constantinople. Historians since Gibbon have been lying and calling it “the Byazantine Empire” because they want to dispossess the later Christian phase of its Romanitas so they can claim Rome for secularism). This is why the marriage service is the most recently composed of all Orthodox Christian services (unless you count akathists and canons to saints whose glorification was proclaimed since then as “new services”. I don’t, I regard them as additions to small compline or variations on a paraklesis or molieben.)
Of course, all Oklahoma will do with this is to oblige homosexuals to have Unitarian-Universalist or Wiccan ceremonies, and to have the law eventually struck down on behalf of atheists who want a way of marrying without involvement of a clergyman of any sort.
I have been trying to tell this to people all along, especially the libertarians who use it as cover that we just make marriage religious.
It is a ridiculous idea on all levels, so ridiculous, that it is just phony nonsense.