Depends on the purpose and intent of its establishment. A lot of the African-American scholarships were set up as an attempt to undue the historical repression and disenfranchisement of Blacks in America.
If the White Scholarship were designed with some notion of racial purity or prejudices towards non-white, then fuck no.
Blacks were singularly united in their collective experiences of being enslaved in the United States and, afterwards, having raised families in poverty in the Jim Crow South. There is little to unite “white” asides from their historical discrimination of minorities– whether it be through the Chinese Exclusion Act, Segregation, or the Japanese internment camps. I can understand scholarships for Irish-Americans, German-Americans, British-Americans, and French-Americans– groups with distinct identity and history in the United States; but definitely not as monolithically and discriminatorily “white.”