The case in question is Hollingsworth v. Perry (2013) which ended up being a key court decision allowing for same-sex marriage in California and is closely related to Obergefell v. Hodges (2015). The Supreme Court of the United States (SCOTUS) over turned California Proposition 8 (Prop 8) which defined marriage as being solely between a man and a woman.
When the case went before the SCOTUS a number of amicus curiae briefings were filed that argued, in essence, marriage is traditionally limited to opposite-sex couples because the government has vested interest in procreation. The Family Research Council filed one example of this arguing,
The Court has recognized a substantive due process right to marry.
Loving v. Virginia, 388 U.S.1 (1967), Zablocki v. Redhail, 434 U.S. 374 (1978), and Turner v. Safley, 482 U.S. 78 (1987). But the
right recognized in these decisions all concerned opposite-sex, not
same-sex, couples. Loving, 388 U.S. at 12, Zablocki, 434 U.S. at
384, Turner, 482 U.S. at 94-97. That the right to marry is limited
to opposite-sex couples is clearly implied in a series of cases
relating marriage to procreation and child rearing.
See Skinner v. Oklahoma ex rel. Williamson, 316 U.S. 535, 541 (1942) “Marriage and procreation are fundamental to the very existence and
survival of the race”); Meyer v. Nebraska, 262 U.S. 390, 399 (1923)
(liberty language in Due Process Clause includes “the right of the
individual . . . to marry, establish a home and bring up children”);
Maynard v. Hill, 125 U.S. 190, 211 (1888) (referring to marriage as “the foundation of the family and of society, without which there
would be neither civilization nor progress”).
This argument first appeared in lower courts where expert witness Nancy Cott noted that "marriage has never been universally defined as a union of one man and one woman, and that religion has never had any bearing on the legality of a marriage" (transcript p.181-210). Legal interpretation of the argument also noted that the position was weak since Prop 8 was underinclusive since jurisprudence would require that it apply to opposite-sex couples that could not procreate. This position was echoed by editorials as well.
This point was then argued before the SCOTUS by Mr. Cooper on behalf of the Prop 8 supporters. The transcripts of the argument indicate that that the justices were skeptical, noting that opposite-sex couples that are incapable of procreation are still permitted to marry, for example,
JUSTICE KAGAN: Well, suppose a State said, Mr. Cooper, suppose a
State said that, Because we think that the focus of marriage really
should be on procreation, we are not going to give marriage licenses
anymore to any couple where both people are over the age of 55. Would
that be constitutional?
MR. COOPER: No, Your Honor, it would not be constitutional.
JUSTICE KAGAN: Because that's the same State interest, I would think,
you know. If you are over the age of 55, you don't help us serve the
Government's interest in regulating procreation through marriage. So
why is that different?
MR. COOPER: Your Honor, even with respect to couples over the age of
55, it is very rare that both couples — both parties to the couple
are, and the traditional — (Laughter.)
JUSTICE KAGAN: No, really, because if the couple — I can just assure
you, if both the woman and the man are over the age of 55, there are
not a lot of children coming out of that marriage.
MR. COOPER: Your Honor, society's — society's interest in responsible
procreation isn't just with respect to the procreative capacities of
the couple itself. The marital norm, which imposes the obligations of
fidelity and monogamy, Your Honor, advances the interests in
responsible procreation by making it more likely that neither party,
including the fertile party to that —
JUSTICE KAGAN: Actually, I'm not even —
JUSTICE SCALIA: I suppose we could have a questionnaire at the
marriage desk when people come in to get the marriage — you know, Are
you fertile or are you not fertile?
The difficulty in answering this question lies in the fact that the SCOTUS decided the case due to lack of standing by the petitioners as opposed to the merits of the case. Despite this it appears that the justices were highly skeptical of the argument as indicated by their questioning in the transcript.
The later Obergefell v. Hodges case actually gives us some context in which to interpret the SCOTUS decision though since the same argument was made. In the slip opinion in which the majority noted that,
The respondents also argue allowing same-sex couples to wed will
harm marriage as an institution by leading to fewer opposite-sex
marriages. This may occur, the respondents contend, because
licensing same-sex marriage severs the connection between
natural procreation and marriage. That argument, however, rests on
a counterintuitive view of opposite-sex couple’s decisionmaking
processes regarding marriage and parenthood. Decisions about whether to marry and raise children are based on many
personal, romantic, and practical considerations; and it is
unrealistic to conclude that an opposite-sex couple would
choose not to marry simply because same-sex couples may do so.
See Kitchen v. Herbert, 755 F. 3d 1193, 1223 (CA10 2014) (“[I]t
is wholly illogical to believe that state recognition of the
love and commitment between same-sex couples will alter the most
intimate and personal decisions of opposite-sex couples”). The
respondents have not shown a foundation for the conclusion
that allowing same-sex marriage will cause the harmful
outcomes they describe. Indeed, with respect to this
asserted basis for excluding same-sex couples from the right
to marry, it is appropriate to observe these cases involve
only the rights of two consenting adults whose marriages
would pose no risk of harm to themselves or third parties.
...thus noting the legal coupling between marriage and procreation, although arguing that procreation cannot be the sole basis for marriage since opposite-sex couples have "many personal, romantic, and practical considerations." The dissenting argument by Justice Roberts also notes that,
The premises supporting this concept of marriage are so fundamental
that they rarely require articulation. The human race must
procreate to survive. Procreation occurs through sexual relations
between a man and a woman. When sexual relations result in
the conception of a child, that child’s prospects are
generally better if the mother and father stay together rather
than going their separate ways. Therefore, for the good of
children and society, sexual relations that can lead to
procreation should occur only between a man and a woman
committed to a lasting bond.
Further noting that,
This singular understanding of marriage has prevailed in the
United States throughout our history. The majority accepts that at
“the time of the Nation’s founding [marriage] was understood
to be a voluntary contract between a man and a woman.”
Ante, at 6. Early Americans drew heavily on legal scholars like William Blackstone, who regarded marriage between “husband and
wife” as one of the “great relations in private life,” and
philosophers like John Locke, who described marriage as “a voluntary
compact between man and woman” centered on “its chief end,
procreation” and the “nourishment and support” of children. 1 W. Blackstone, Commentaries *410; J. Locke, Second Treatise
of Civil Government §§78–79, p. 39 (J. Gough ed. 1947). To those who drafted and ratified the Constitution, this
conception of marriage and family “was a given: its structure, its
stability, roles, and values accepted by all.” Forte, The
Framers’ Idea of Marriage and Family, in The Meaning of
Marriage 100, 102 (R. George & J. Elshtain eds. 2006).
In a similar dissenting argument, Justice Alito notes that,
If this traditional understanding of the purpose of marriage does not
ring true to all ears today, that is probably because the tie
between marriage and procreation has frayed.
As such, we may conclude that the justices accept as legal fact one of the purposes of marriage is to legitimize children that result from a couple. However, the majority decision in Obergefell v. Hodges rejects the legal argument that the sole legal and historical reason for marriage in the United States is to regulate procreation.