How marriage occurs with non Jews, Hilchos Ishus 1:1
קֹדֶם מַתַּן תּוֹרָה הָיָה אָדָם פּוֹגֵעַ אִשָּׁה בַּשּׁוּק אִם רָצָה הוּא וְהִיא לִשָּׂא אוֹתָהּ מַכְנִיסָהּ לְתוֹךְ בֵּיתוֹ וּבוֹעֲלָהּ בֵּינוֹ לְבֵין עַצְמוֹ וְתִהְיֶה לוֹ לְאִשָּׁה.”:
Before the giving of the Torah, it would be that if a man happened upon a woman in the marketplace and they wanted to marry each other, he would bring her into his house and consummate the marriage between them privately, and she would be his wife.
How divorce works with non jews Hilchos Melachim 9:8
וּמֵאֵימָתַי תִּהְיֶה אֵשֶׁת חֲבֵרוֹ כִּגְרוּשָׁה שֶׁלָּנוּ מִשֶּׁיּוֹצִיאֶנָּה מִבֵּיתוֹ וִישַׁלְּחֶנָּה לְעַצְמָהּ. אוֹ מִשֶּׁתֵּצֵא הִיא מִתַּחַת רְשׁוּתוֹ וַתֵּלֶךְ לָהּ. שֶׁאֵין לָהֶם גֵּרוּשִׁין בִּכְתָב. וְאֵין הַדָּבָר תָּלוּיִ בּוֹ לְבַד. אֶלָּא כָּל זְמַן שֶׁיִּרְצֶה הוּא אוֹ הִיא לִפְרשׁ זֶה מִזֶּה פּוֹרְשִׁין:
And when is one of their married women considered like one of our divorcees? From the time he expels her from his house and sends her off on her own, or from when she leaves on her own from his property. For the Gentiles do not have a written bill of divorce. and the matter is not dependent upon him alone. Rather, whenever either he or she wishes to separate, they (may) separate.
What the non Jewish courts do, ibid halacha 14
וְכֵיצַד מְצֻוִּין הֵן עַל הַדִּינִין. חַיָּבִין לְהוֹשִׁיב דַּיָּנִין וְשׁוֹפְטִים בְּכָל פֶּלֶךְ וּפֶלֶךְ לָדוּן בְּשֵׁשׁ מִצְוֹת אֵלּוּ. וּלְהַזְהִיר אֶת הָעָם.
What must they do to fulfill their requirement regarding the Law of Justice? They have to set up magistrates and judges in each district to judge the people with regard to these Six Commandments; and they must issue warnings (about them) to the people.
The Kesef Mishnah there brings another peshat, but no mention that they can decide what constitutes marriage. Sounds pretty objective.