As an Indian, I don't think we have an obsession with white women. If anything, the majority of India have an obsession with their cultural values from the place they come from. Also yes, sexism is a gigantic problem within a majority of Indian societies. There is a clear divide between the ideologies of the current young adult generation of Indians compared to the generations of their fathers and mothers(note, not applicable to every Indian family obviously).
HOWEVER, there is this thing where 'fairer' skin is looked at as more as being more attractive. Historically, you can attribute that to the kind of thinking towards an idea where people with clearer or fairer skin come from wealthy and respected families. Everyone knows the Mughals bathed in milk to get clearer and whiter skin.
Rural India, still respects the ancient and archaic caste system we have, in the sense that now, castes are seen as communities rather than just nobility. Marriage for a majority of Indians is decided by the family rather than the one who wants to be married. In India, marriages are have a bigger place in society than arguably any other place in the world. It's a chance to earn more money, respect, power, a very vague sense nobility and above all, see that their family name is not tarnished in front of their communities.
HOWEVER! While people want to remain in their own communities, A dalit(essentially the lowest rung of the caste system) would much rather marry into a brahmin(the highest in the caste system) family rather than stay a dalit. Brahmins are typically considered to have fair skin. So there you go. So basically, it's all about maintaining an image in front of others. Fairer skin is considered to be a trait of nobility, wealth and the like. Image and power projection is almost everything in Indian culture despite what people may have you believe.
For example: I come from a Muslim family from a state in India called Kerala. People of Kerala are called Malayali's. My parents are pretty open, they know I drink, smoke occasionally and have a fair-skinned Tamil Hindu girlfriend for years now (only my mom knows I'm agnostic though). However, my parents want me to marry a muslim. HOWEVER, not just any muslim, but a muslim from Kerala and not just any muslim from Kerala, but a family that's within our community or very well known to our community within Kerala(still the best parents in the world tho'). In India, the idea is that families marry, not the bride or groom. Same is true for a good chunk of Indian families out there.
Tl;dr : It's not entirely racism, it's more about stereotypical perceptions of class, wealth and beauty that's heavily ingrained in our culture. For if you have all that, your skin color really doesn't matter. That being said, I know a lot more 'hotter' men and women of darker skinned Indians than I do of fairer skinned Indians. And well, Indians, when they're hot, they're some of the hottest men or women you will ever see in your life. (Silly Patriotism ftw!)
I know it's too long and could be boring, but I hope that answers your question.