The Bible does not specifically prohibit interracial marriage. Christians are free to marry whomever they wish providing it is a believer (1 Cor. 7:39; 2 Cor. 6:14).
The Old Testament gave some restrictions on whom the Israelites could marry. The other people groups were idolaters and worshippers of false gods. They were to be protected from pagan influences (Deut. 7:3-4) and to guard the purity of the Aaronic priesthood (Lev. 21:13-14). The various tribes were not to be diluted by intermarriage with unbelievers (Num. 36:3-9).
Race is not used in the Bible. We all came from “one blood” from Adam (Acts 17:26) and uses the term “nations” or people groups. Revelation 14:6 refers to them as “every nation, kindred, tongue, and people). But for the sake of the question, I will use the term “race.”
Interracial marriage, like all marriages, should be considered only after much prayer, discernment, and wisdom. There is a stigma still attached today that can cause great harm for the adults, children, and family members. The children can be greatly affected by differences of culture that can hinder acceptance by other children and adult family members.
Paul said that all things are lawful, but all things are not expedient. Sometimes the Scriptures may not prohibit certain things, but we must consider other factors when making these decisions (Rom. 14-15).