Do scriptures allow Hindus to marry foreigners, or non-Hindus of other races?
Inter-caste marriage has its restrictions, but what about interracial marriages?
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No, the Dharma Shastras explicitly forbid Hindus from marrying foreigners, otherwise known as Mlecchas.
Here are some verses from the Atri Smriti that prohibit interracial marriages:
The purification for knowing [having sex with] a Mleccha woman consists in a Santapanam penance, or it is said, one may attain to purification by performing a Taptakrcchra penance.
If a person lives with a wife known [who has had sex with] by a Mleccha man, he becomes purified by bathing with the cloth on and drinking clarified butter.
By cohabiting [living with and having sex with], unknowingly, with the women of the Chandalas, Mlecchas ... one is purified by a Paraka.
But if one knows them willingly, and procreates children with them, then he is degraded to the same caste as them; there is no doubt about it, for that man is born as her son.
So it is actually a huge sin to marry foreigners, since one becomes degraded to the caste of Mleccha, and there seems to be no expiation for knowingly having sex with a Mleccha.
This answer explains some of the different types of penances.