The Bible does not forbid interracial marriages. It does, however, forbid a Christian from marrying an unbeliever: “Do not be bound together with unbelievers; for what partnership have righteousness and lawlessness, or what fellowship has light with darkness?” (2 Cor. 6:14).
In Numbers 12:1-8 is an account of Moses marrying an Ethiopian woman. “Then Miriam and Aaron spoke against Moses because of the Cushite woman whom he had married (for he had married a Cushite woman),” (Num. 12:1).
Other Bible verses translate this as “Cush.” Cush is the ancient location of Ethiopia. Miriam and Aaron were upset with Moses because he married a Cushite woman. Cush has also been considered to be the northern part of Egypt. Either way, it is very possible that the Cushite woman was of a different race. God defended Moses for marrying this woman. If race were an issue, this would have been a great place to set the matter straight.
Also, people of different races were not mentioned as being among those forbidden by God for Jews to marry, see Exodus 34:11.