American Independence Day is a national holiday celebrated annually on 4 July to commemorate the adoption of the Declaration of Independence in 1776. It is commonly referred to as the birthday of the United States of America.
Harpo's statement that this is a holiday for white people highlights the fact that African-Americans at that time did not gain 'independence': the battle began for them with the birth of the modern African-American Civil Rights Movement in 1955.
Alice Walker herself is still a campaigner for civil rights. Her speech 'Activism is the Rent I Pay for Living on this Planet', given at Evening of Conscience, 2 October 2006, can be heard below.