That comes and goes, most Americans who bring that up, are thinking about the relatively modern era of when the Catholic church was the authority, or some state religion, which is not something available to Americans, and not what most of us think would be preferable to us passing our own laws.
Thomas Jefferson, who was involved in marriage/divorce law, did not want the Pope running marriage law and didn't like that they had taken it over from Rome when the Catholic church became powerful enough.
“...thinking about the relatively modern era”
Yes, and that makes for a very shallow understanding.
The first hunter-gatherer and farming states had and enforced marriages.
A well-known example of one reason they had to do so is in the story of David and Bathsheba.