Absolutely not. No one can be denied service in any place of public accommodation, including restaurants, barber shops, car dealerships, etc., because of race. Anyone so discriminated against may file a lawsuit under the Civil Rights Act of 1964, the purpose of which was to eliminate the inconvenience, unfairness and humiliation of racial discrimination. The exact statute is 42 U.S.C. Section 1981(b), which states that all persons within the jurisdiction of the United States shall have the same right in every state and territory to make and enforce contracts and to the full and equal benefit of all laws and proceedings for the security of persons and property as is enjoyed by white citizens. Refusing to contract with or serve someone on the basis of their race amounts to a per se violation of 42 U.S.C. Section 1981(b).
If you have been discriminated against, you may bring a lawsuit for damages. To initially prove a racial discrimination case, you must show that you were a member of a racial minority, that the defendant (restaurant, barbershop, or car dealership owner or employee) and had the intent to discriminate on the basis of race.