The first thing to note is that your question is kind of the wrong way around. US states are sovereign and generally have the ability to make any kind of laws they want, unless they violate some specific tenet of federal law or the US Constitution.
You suggest, for instance, that arguments which are "culturally founded" have no place in the law, but that's just your opinion, and there isn't generally anything preventing a state from making law based on such things, should its elected legislature see fit to do so. (Indeed, one could argue that nearly all laws are in some sense "culturally founded", since they are based on some notion of what kind of behavior is or is not appropriate, and those tend to be culturally based.)
So legally speaking, the states aren't, by default, obligated to give any sort of justification for the laws they made. The burden of proof is on the other side. Someone seeking to overturn those laws would have to convince a court that the laws violated some specific provision of the Constitution (or another superior law). If they couldn't convince a court of this, the law would stand.
From what I have read, before the US Supreme Court's 2015 legalization of same-sex marriage in Obergefell v. Hodges, the previous precedent was set in 1971 by the Minnesota Supreme Court in Baker v. Nelson. The decision itself is quite short and is worthwhile to read. Quoting Wikipedia's summary, the plaintiffs claimed that Minnesota's restriction of marriage to opposite-sex couples violated several provisions of the US Constitution:
First Amendment (freedom of speech and of association),
Eighth Amendment (cruel and unusual punishment),
Ninth Amendment (unenumerated right to privacy), and
Fourteenth Amendment (fundamental right to marry under the Due Process Clause and sex discrimination contrary to the Equal Protection
Clause).
The Minnesota court determined that none of the plaintiffs' objections were valid. Again, I'll refer you to the decision for the details, but the court mainly focused on their Fourteenth Amendment arguments (the others may have been addressed by the trial court, whose opinion I can't find online). They wrote:
The equal protection clause of the Fourteenth Amendment, like the due process clause, is not offended by the state's classification of persons authorized to marry. There is no irrational or invidious discrimination.
They specifically rejected any analogy to bans on interracial marriage, which had been held unconstitutional in Loving v. Virginia:
But in commonsense and in a constitutional sense, there is a clear distinction between a marital restriction based merely upon race and one based upon the fundamental difference in sex.
Baker appealed to the US Supreme Court, but his appeal was dismissed "for want of a substantial federal question," without any further explanation. (Nobody quite seems to understand what they meant by that, but here is an essay discussing the situation in a little more depth.) The effect of the dismissal was that the Minnesota court's decision became binding precedent upon the whole nation - laws against same-sex marriage didn't violate those provisions of the Constitution. And that was how matters stood for 44 years until Obergefell. (Of course, there was nothing to stop individual states from deciding to allow same-sex marriage, and some in fact did so in the meantime.)
You have suggested that laws against same-sex marriage were religiously motivated. This might suggest an argument that they would violate the Establishment Clause of the First Amendment. The plaintiffs in Baker didn't raise that point, so it wasn't considered in the Minnesota court's opinion. I don't know whether any other courts have considered it; no such argument was mentioned in the opinion in Obergefell.
