There's no state interest in denying this.
Following the
Nevada doctrine, Utah conservatives seem to be determined to make their anti-marriage-equality case
as silly as possible.
In their opening brief in support of Utah’s appeal of a ruling that the state’s ban on same-sex marriages is unconstitutional, the Beehive State’s lawyers argued Monday that one reason to avoid marriage equality was the “correlation between genderless marriage and lower birthrates.” [...]
It is also striking that fertility and birthrates tend to be markedly lower in nations and states that have embraced same-sex marriage. For example, the birthrate in states (and Washington, D.C.) that have adopted a genderless marriage definition is significantly lower than the national average. In fact, the six lowest birthrate states have all adopted that redefinition, while none of the nine highest birthrate states have done so.
In a footnote, the attorneys reference the Centers for Disease Control’s National Vital Statistics Reports – Births: Final Data for 2012, which identified the six states with the lowest birthrates in that year were Connecticut (10.2 live births per 1,000 estimated population), Maine (9.6), Massachusetts (10.9), New Hampshire (9.4), Rhode Island (10.4), and Vermont (9.6). The states with highest birthrates in that report Texas (14.7) and Utah (18.0).
You may immediately see the problem with this, because you are not a Utah conservative. Correlation is not causation; you cannot say that certain states had lower birthrates in 2012
because they are more supportive of marriage equality. (Think Progress also notes that the same birthrate statistics held true in 2000, before marriage equality had the widespread support it currently enjoys. Go figure.) Lower birthrates can be more directly correlated with things like, say, higher educational status or higher income or religious beliefs or better health care options, all of which might have considerably more to do with the northeast continuing to have lower birthrates than stack-em-like-cordwood Utah. You might as well go whole hog and declare that smart people have fewer children, though Utah conservatives obviously do not want to phrase it like
that.
The other flaw in the argument would be the apparent supposition that denying marriage equality will make gay Americans go out and have more babies, which only makes sense if you do not understand how these things work.
A question, though: Why do state conservatives consider a high birthrate to be inherently and obviously more desirable than a low one? The point being ... what, now? America is in no danger of being underpopulated, not even in the slightest, and there would seem to be an upper bound to how many people the nation can reasonably hold (again, without resorting to cordwood techniques.) Is this a religious thing? A bar bet among the states? The complex beginnings of a future real estate scam? Even if you were to brush aside the rather glaring problem of the Utah conservative lawyer class being statistically illiterate and/or not understanding the basics of human reproduction, even the assertion that state interests require marriage equality to be denied seem to lack clarity on what, precisely, those state interests would be.