Discrimination against skin tone for women of different cultures is fairly common. However, I think it is simplistic and American-centric to assume that this is a white people-black people issue.
I suspect in many cases it is a class thing. For example, I'm Chinese American and as a kid my mother was always harping on me to wear sunscreen because only "peasant women who work in the fields look that black." (Class relations in the United States wasn't even on her radar.) When travelling through Vietnam, I realized that even peasant women covered themselves head to toe in stifling heat to block the sun. The palest people were the ones who worked in shops and office buildings. The darkest people were the poorest of the poor, the ones with no land or home, so they could not escape the sun.
In Cambodia, a temple guide shocked me by casually mentioning how no Cambodian could understand why the white French colonialists had seemed so attracted to the darkest skinned farm girls in the countryside, because for Cambodians the ideal of beauty was the pale skinned court dancer. I got the distinct impression she thought the French were stupid.
Finally, an Indian friend once told me that fair skin is prized, because dark skin indicated low caste, or even worse, untouchable. Which he openly admitted was unfair, because untouchables often had the crappiest jobs and the worst living conditions---stuck out in the sun.