Answer by Balaji Viswanathan:
[I'm going to take a harsh tone for this answer, as this same question keeps popping up and I am tired of giving nice answers for this. See my last answer:]
The central idea of secession is anarchy
— Abraham Lincoln
Self-determination of peoples – a phrase. . . simply loaded with dynamite
— Robert Lansing (former secretary of State under Wilson)
Nations are not the irreducible unit of political identity. Within a nation there are regions, provinces, tribes, faiths, factions, clans. And then it's every man for himself. 
The Crimea referendum was a farce that Russia enacted after it took over. The Scottish and Quebec referendums have failed to achieve independence despite promising a lot. Most such referendums are stupid and pointless. Here is how.
Jammu & Kashmir is not a monolithic entity. It is comprised of 3 regions – Hindu Jammu, Buddhist Ladakh, and Muslim Kashmir. These 3 regions would likely vote differently. Why should Jammu follow Kashmir valley out of India [if Kashmir valley votes for independence]? If the state of Jammu & Kashmir votes out of India through a referendum, then logically Jammu & Ladakh need to have separate referendums to ask if they want to be separate, join the rest of India or stay with Kashmir.
But then Jammu, Kashmir, and Ladakh are not monolithic entities either. Some Muslim towns of Jammu might want to stay with Kashmir, while some Hindu villages of Kashmir might want to move where Jammu & India heads.
You can infinitely keep reducing this process, until you have no more to divide.
I can go to my rooftop and want a separate kingdom of Balaji that includes just my house. The government can conduct a referendum with the people of my house and they might vote with me. That's it. My vote would make my home a new kingdom separate from India and I don't need the input of any other Indian in this decision. Would you call that absurd? Yes. The same is true of all the referendums above.
[This is not even a theoretical worry, but is a current event. In 2011, South Sudan won a separate country through a referendum. Western countries that broke Sudan this way were patting their own backs before the reality came. Now, it is in the midst of a major pandemonium. The Nuer people there don't want to stay with the rest of South Sudan & are leading a major armed struggle. Will they get their own North-east South Sudan? At what point does such a division end?]
Why India will never let go of Jammu & Kashmir?
- Partitions are brutal. India has suffered enormously in 1947. Millions were displaced & a part of them died.
- The Indian Constitution was built with the idea that people of diverse languages, cultures and religions can live in harmony. If Kashmir breaks away, the fundamental assumption is broken. Soon, there will be anarchy all over India.
- People of different religions will ask for their own nations [Khalistan modeled after the earlier formation of Pakistan], then different ethnicities will ask for their own [like what is happening with ULFA and others in Northeast] and different economic classes [like what Naxals are doing & Dalitistan groups] too. The problem is we all have multiple identities and these claims are overlapping. Classic Pandemonium with a million civil wars.
- Kashmir's breakaway could break the Indian union and we might see a sea of new states warring each other. Students of Indian history know what happens to the subcontinent when it is that way. India's greatness always came when the subcontinent was unified – liked under the Mauryas, the Guptas, and later the Mughals.
- The Indian subcontinent was geographically meant to be a single logical unit. That would make all the rivers originate and empty within its own borders and provide adequate natural borders. Splitting this arrangement does terrible things for security and water sharing.
In short, there is no logical way India will give up Kashmir. India's existence depends on it and the amputation of its head [in terms of geography] will happen iff someone more powerful can defeat the Indian army in its own home field. There exist no such powers now. In fact, other than Pakistan no one even pays attention to the issue any more.
Further reading on this: