I think you’re inverting my position. the government can delegate its authority to officiate at a wedding, but it cannot refuse to recognize a Catholic (Lutheran/Jewish/Islamic, etc.) marriage. I’m presuming the validity of the religious service.
We may be talking about two different issues. They do recognize a wedding which was performed legally in another state or country. However, an ordained clergyperson cannot necessarily simply show up and perform a legally binding ceremony anywhere without fulfilling the legal requirements of the state in which the marriage is to take place, including restrictions on whether s/he has to register in advance, and following the laws on filling out and returning the marriage license.
Restrictions are most often related to clergy who do not reside in the state in which they are to perform the marriage, but some states require all clergy to obtain a license to perform (different than the marriage license) or otherwise register. There are also issues with marriages performed in foreign countries (
ehow.com/how_7961_find-foreign-marriage.html)
So, no, just the fact that an ordained clergyperson performed a sacramentally valid marriage does not automatically make it a legally valid one according to US law. Just because Uncle Joe is a Catholic priest in another state doesn’t necessarily mean he can marry you legally in your state without jumping through some hoops and if he fails to file the marriage certificate within the specific state guidelines (which range from less than a week to a month or more), there could also be problems.
usmarriagelaws.com/search/united_states/index.shtml
Some examples (not comprehensive):
Arkansas: Ministers must have their ordination credentials filed by the county clerk who will then issue a certificate to the minister.
Hawaii: Ministers must obtain a license from the department of health before performing marriages.
Kansas: Ministers are required to file credentials or ordination with the judge of a probate court before performing marriages.
Massachusetts: Out-of-state clergy need to obtain a Certificate of Authorization from the Massachusetts Secretary of the Commonwealth before the wedding ceremony.
Kentucky: Ministers must be licensed before performing marriages.
Louisiana: Ministers must register with the clerk of the district court of the parish or with the health department if in New Orleans.
Massachusetts: Before performing marriages, ministers are required to apply for a certificate from the state. You must file a copy of your ordination certificate and a statement from the church saying that you are in good standing.
Missouri: A clergyman or clergywoman, active or retired, who is in good standing with any church or synagogue in Missouri; A religious society, institution or organization in Missouri of which either marriage party is a member, in accordance with the organization’s regulations and customs.
Nevada: Ministers are required to complete a complicated procedure to obtain a certificate of permissions to perform marriages. Among other requirements, the applicant’s ministry must be primarily one of service to his congregation or denomination and his performance of marriages must be incidental to such service.
New Hampshire: An ordained clergyman, who is not a resident of New Hampshire must obtain a special license which can be used only for that particular marriage;
Ohio: Before performing a marriage, ministers must present their ordination credentials to the probate judge of any county. The judge will provide the minister with a license to perform marriages. The minister must then present his license to the probate judge in any county in which he performs a marriage.
Oklahoma: Ordained ministers of the gospel of any denomination who are at least 18 years of age may perform marriages. Ministers must file a copy of their credentials with the county clerk before performing marriages.
Oregon and Pennsylvania: Ministers of any church organized, carrying on its work, and having congregations in this state may perform marriages in this state if authorized by their church to do so. Before performing marriages, ministers must file their credentials with the county clerk of the county in which they reside or in which the marriage is to be performed.
Rhode Island: Ministers must obtain a license from the city or town clerk before performing marriages.
Vermont and Virginia: Ordained ministers residing in this state may perform marriages. Non-resident ordained ministers may perform marriages with the permission of the probate court of the district within which the marriage is to take place.