Here's a more cut and dry answer.
Talmud Bavli, Chulin 92a--b:
עולא אמר אלו שלשים מצות שקבלו עליהם בני נח ואין מקיימין אלא שלשה אחת
שאין כותבין כתובה לזכרים ואחת שאין שוקלין בשר המת במקולין ואחת שמכבדין את התורה
Said Ulla: There were thirty commandments that Noahides accepted, but they keep only three: not to write a ketuba for males; not to weigh dead human flesh in the market; and to show respect for the Torah.
Noahide-ism recognizes marriage between a man and a woman (as whatever ceremony that culture considers normative, it need not look like kiddushin); cheating on that marriage would constitute adultery. It does not respect it between two men, two women, siblings from a common mother, or several other very close relatives. See Rambam, Laws of Kings and their Wars, Chapter 9.
Joel, not sure if this point was confusing you: in Noahidism (i.e. Judaism's take on what non-Jews should keep), marriage and divorce are whatever society deems them to be. So New York State's civil marriage and divorce are 100% fine for non-Jews, according to Judaism. Similarly Judaism has no problem with non-Jews eating as much pork as they like. But for a country to officially call something marriage that Noahidism says could never be marriage seems contrary to the value system of Noahidism; hence many Jews find it troubling.
As for how much/little Talmudic mention it gets, again for Jews this was all a moot point, no kiddushin, nothing to talk about. The objection here is related to Noahide law, which only occupies a few pages of the Talmud; in that context, the quotes above are not "scraping."
All the discussions of mamzer and civil marriage/divorce are with regards to Jews, which is a distraction from the question at hand, which is how Noahidism should influence a policy by non-Jews, for non-Jews. (By the way "bastard" is a poor translation; a child born out-of-wedlock is 100% kosher [though people may still point fingers ...]. A mamzer is a product of adultery or incest.)
Lastly there is the distinction between things that are simply not prohibited, versus being actively recognized as official. Rabbi Isaac Arama of Spain (1400s), in his Akeidat Yitzchak (Vayeira #20), sadly noted that yes, some Jewish "bachelors and idiots" will get in trouble with women they shouldn't. But for rabbis to provide communal funds (as some in his time did) for all-Jewish houses-of-ill-repute (as much as it would prevent some problems) would be to institutionalize a transgression, which is beyond the pale. He compares it to Sodom, not in the Christian reading that it was all about sex (of whatever flavor), but by the Talmudic reading that it was all about corruption, power, and selfishness (raping visitors wasn't about lust, it was about showing who's in control); where those problems were not simply overlooked, they became the law.
(Devil's advocate once again, as Rabbi Michael Broyde has observed, US Law allows things that Noahide law doesn't. The same religious protections that let me keep Judaism also let someone worship idols; that's a trade-off we accept. Some have argued that should be the case here too. But you'd asked to understand the objection.)