Does a non-Jew have an obligation to get married like the Jew does?
Rabbi Achai Gaon, in Shiltos (165) states that non-Jews are obligated to marry and have children.
See also Rabbi Moshe Feinstein in Igros Moshe (E"H book 4 ch. 29) where he concludes that the obligation of Pru Urvu (be fruitful and multiply) is an obligation on all of mankind, as a communal commandment. This also seems to be the opinion of Drisha (E"H ch. 1 note 2). It is the individual mandate to have children that was only given to Jews.
See also Yevamos (62a) where apparently Rabbi Yochanan held that one fulfills his obligation with non-Jewish children he had before his conversion to Judaism. (Rambam however adds that those children must have also converted.)
(UPDATE: on reflection, since non-Jews do not have the ability to effect a legally binding Kidushin/Nisuin, they couldn't be obligated to marry, at least not in the same sense as a Jew.
Noahide-law adultery involves not a married woman, but a woman who is living together with a man without the formal institution of marriage, aka "be'ulas ba'al".)
Apparently not. Rambam (Laws of Marriage 1:1,4) states that Torah law (for Jews) requires a marriage to be effected in the presence of witnesses, and outlaws non-marital sex. But neither of these applies to non-Jews.
Nor would they be required to marry (or have a non-marital sexual relationship) in order to have children. The Gemara (Sanhedrin 59b) states that the mitzvah of "be fruitful and multiply" (Gen. 1:28) is one of the ones that was not repeated at Sinai, meaning that only Jews are obligated in it.
(To be precise, the Gemara says that it was repeated - in Deut. 5:26, by implication - but that this was done in order to teach us how halachic enactments have to be repealed, rather than to make it binding on non-Jews. Presumably, the She'iltos (cited in Barry Frank's answer) would understand this Gemara differently: this mitzvah is repeated for two reasons - to make it obligatory on non-Jews as well as to teach us about the rules of halachic repeal.)