I don't think I can argue better than Obergefell v. Hodges itself where they cite Loving as precedent:

http://www.supremecourt.gov/opinions/14pdf/14-556_3204.pdf

>Applying these established tenets, the Court has long
held the right to marry is protected by the Constitution.
In
Loving
v.
Virginia
, 388 U. S. 1, 12 (1967), which invali
-
dated bans on interracial unions, a unanimous Court held
marriage is “one of the vital personal rights essential to
the orderly pursuit of happiness by free men.” The Court
reaffirmed that holding in
Zablocki
v.
Redhail
, 434 U. S.
374, 384 (1978), which held the right to marry was bur
-
dened by a law prohibiting fathers who were behind on
child support from marrying. The Court again applied
this principle in
Turner
v.
Safley
, 482 U. S. 78, 95 (1987),
which held the right to marry was abridged by regulations
limiting the privilege of prison inmates to marry. Over
time and in other contexts, th
e Court has reiterated that
the right to marry is fundamental under the Due Process
Clause. See,
e.g., M. L. B.
v.
S. L. J.
, 519 U. S. 102, 116
(1996);
Cleveland Bd. of Ed.
v.
LaFleur
, 414 U. S. 632,
639–640 (1974);
Griswold
,
supra
, at 486;
Skinner
v.
Okla
-
homa ex rel. Williamson
, 316 U. S. 535, 541 (1942);
Meyer
v.
Nebraska
, 262 U. S. 390, 399 (1923).

>A first premise of the Court’s relevant precedents is that
the right to personal choice
regarding marriage is inherent
in the concept of individual autonomy. This abiding con
-
nection between marriage and liberty is why
Loving
inval
-
idated interracial marriage bans under the Due Process
Clause. See 388 U. S., at 12; see also
Zablocki
,
supra,
at
384 (observing
Loving
held “the right to marry is of fun
-
damental importance for all individuals”). Like choices
concerning contraception, family relationships, procrea
-
tion, and childrearing, all of which are protected by the
Constitution, decisions concerning marriage are among
the most intimate that an individual can make. See
Law
-
rence
,
supra
, at 574.

>ee App. to Brief for Appellant in
Reed
v.
Reed
, O. T.
1971, No. 70–4, pp. 69–88 (an extensive reference to laws
extant as of 1971 treating women as unequal to men in
marriage). These classifications denied the equal dignity
of men and women. One State’s law, for example, pro-
vided in 1971 that “the husband is the head of the family
and the wife is subject to him; her legal civil existence is
merged in the husband, except so far as the law recognizes
her separately, either for he
r own protection, or for her
benefit.” Ga. Code Ann. §53–501 (1935). Responding to a
new awareness, the Court invoked equal protection prin
-
ciples to invalidate laws imposing sex-based inequality on
marriage. See,
e.g., Kirchberg
v.
Feenstra
, 450 U. S. 455
(1981);
Wengler
v.
Druggists Mut. Ins. Co.
, 446 U. S. 142
(1980);
Califano
v.
Westcott
, 443 U. S. 76 (1979);
Orr
v.
Orr
, 440 U. S. 268 (1979);
Califano
v.
Goldfarb
, 430 U. S.
199 (1977) (plurality opinion);
Weinberger
v.
Wiesenfeld
,
420 U. S. 636 (1975);
Frontiero
v.
Richardson
, 411 U. S.
677 (1973). Like
Loving
and
Zablocki
, these precedents
show the Equal Protection Clause can help to identify
and correct inequalities in the institution of marriage,
vindicating precepts of liberty and equality under the
Constitution.