The equal protection clause argument was that the ERA was not necessary for that reason--still can't turn a statute on gender because it deprives the other gender of equal protection.
That case got to the U S Supreme Court on precisely this statute. The case was on the other side--the child of a father citizen. So it's not in point on the question presented here.
The child of the father citizen argued that it was not constitutional for the statute to deny him citizenship because if it had been his mother who was a citizen, he might have received citizenship under the statute.
The Court held no citizenship for the child of the father citizen which would have been the result if the statute was not constitutional because of the unconstitutional classification. The majority opinion is fuzz.
There was an extensive scholarly dissent (I forget from who) which said that instead of simply upholding the denial of citizenship to the father citizen's child, the court should have simply ruled the entire statute unconstitutional which it appears to be on its face. Subsequently there is a Cornell law review article quoted or copied into the Long Thread on this topic explaining why the dissent sets forth the law.
On the facts here, it no longer makes any difference because the question of "wedlock" under the statute gets resolved, at least in my view, by the general case law that would find them married for purposes of the statute whether there was a civil event or not.
It further makes no difference because bare citizenship with respect to a child born outside the county is not the issue in any event--it still doesn't free the child from the sovereignty at birth of another government and thus the child isn't a natural born citizen.