No, the Bible teaches that race is unimportant in our relationship with God:
For there is no difference between Jew and Gentile—the same Lord is Lord of all and richly blesses all who call on him. (Romans 10:12 NIV)
The New Testament also makes no mention of race when discussing marriage. It does, however, command that believers should only marry other believers. For example, Paul wrote
A woman is bound to her husband as long as he lives. But if her husband dies, she is free to marry anyone she wishes, but he must belong to the Lord. (1 Corinthians 7:36 NIV)
There was a similar law in the Old Testament: believers were only to marry other believers. However, in the Old Testament believers were almost always Israelites, and so the law was often stated in terms of Israelites not being allowed to marry people of other nationalities. For example:
When the LORD your God brings you into the land you are entering to possess and drives out before you many nations---the Hittites, Girgashites, Amorites, Canaanites, Perizzites, Hivites and Jebusites, seven nations larger and stronger than you ... Do not intermarry with them. Do not give your daughters to their sons or take their daughters for your sons, for they will turn your sons away from following me to serve other gods, and the LORD's anger will burn against you and will quickly destroy you. (Deuteronomy 7:1-4)
The verses make it clear that it was not their nationality that was the problem, it was their religion. This is evident when Boaz married Ruth, a Moabite woman (Ruth 4:10,13). Although it was normally forbidden to marry a Moabite (Neh 13:23-25), in this case it was acceptable because she had converted to belief in the true God (Ruth 1:16).