I think your question can best by answered by two sets of answers.

Firstly, the India of today is not the same as the India in the immediate aftermath of Independence and the Partition. India was comprised of myriad princely states and regional entities. Princely states were given the option of joining either India or Pakistan or staying independent. Broader regional entities (e.g. the United Provinces, today Uttar Pradesh) joined India/Pakistan based on religious composition.

Post Independence leaders of India had to very forcefully incorporate in all of these disparate parts of the whole. Sardar ("Chief") Vallabhai Patel is a national hero today primarily because he was instrumental in convincing various princely states to join the Union of India, either through cash payments to the royal families or through the threat of military force. In some cases, this was a very complex process. Kashmir, for instance, had a Muslim majority but a pro-India king (who actually initially wanted to remain independent). The aftermath of that is, well, beyond the scope of this post. The state of Hyderabad similarly wanted to remain independent, and indeed did so for a couple years after Independence. Only in 1948 did India launch Operation Polo to depose and integrate the state back into the Union.

The point I'm trying to make is that post-Independence India was not politically cohesive. It wasn't some politically integrated whole of which further pieces could be partitioned off; it was a hodgepodge of various regions and princely states that many times had to be forcefully integrated into the 'whole' that is the Union of India.

That leads me to my second point. The Partition was very much a top down initiative. It was the product of backroom deals and politicking by leaders of the Congress (e.g. Nehru, Gandhi, etc.) and the Muslim League (e.g. Jinnah) as well as the British (namely Mountbatten). Numerous other movements did emerge after Independence that were instead bottom up, movements that also called for 'partition' to some degree or another.

On the most extreme end we have the Khalistan movement in Sikh Punjab asking for a separate Sikh-Punjabi state as well as various 'rebellion' movements in the Northeast (e.g. Tripura Liberation Front, etc.). On the more lighter side of things we have ethno-lingual movements in states like Maharasthra and Tamil Nadu.

Why did these 'Partition' movements didn't pan out is pretty much rooted in the Government of India's actions. GoI has not shied away from military force in the most extreme cases, deploying soldiers in the Northeast and broader, much more controversial military operations like Operation Blue Star, which effectively put an end to the Khalistan movement (which was, before the Operation, actually amassing arms for an armed rebellion).

But one could also interpret the GoI's socioeconomic programmes in a similar context. India embraced socialist state planning after Independence, a doctrine whose apex was perhaps PM Indira Gandhi's nationalization of heavy industries, banking, etc. (Coincidentally, PM IG was the same PM who oversaw Operation Blue Star, a decision for which she paid with her life when she was assassinated by her Sikh bodyguards.) Such policies naturally led to the consolidation of economic resources, planning, and power at the Centre- in New Delhi.

At the same time, Indian policymakers pursued various programmes of social consolidation as well. The Constitution empowered the GoI to privilege Hindi as India's national dialect (English as only a convenient remnant of the British Raj to be phased out with Hindi's introduction). Indian leaders like Nehru and Gandhi tried to craft and project an Indian 'identity' abroad. (How these conceptions were very Orientalist and perhaps misguided is worth a post on its own.)

These initiatives had varying success. Regions closer to New Delhi like Rajasthan, for instance, completely embraced Hindi and replaced older languages/dialects in day-to-day use. Regions further from New Delhi like Tamil Nadu completely rejected Hindi, and to this day Tamilians refuse to speak Hindi out of ethnic pride. Regions like Maharashtra pushed back in different ways with the Maratha Nationalist movement using mobs to enforce the use of Marathi in Mumbai and violently harassing southern migrants who were culturally completely different than 'native' Marathis.

I guess all of this is a very long winded way of saying a sort of political framework was crafted and immensely invested into by the Government of India after Independence. This framework involved, sure, but newer 'Partition' movements ultimately had to bow to this framework and the incentives and policies and institutions it nurtured.

The best metaphor I have is being a camp counselor of dozens of kids who are completely different from one another and even resent each other in some cases. In camp counselor India's case, the strategy was to almost 'force' them to bond by locking them in a room and forcefully keeping them in that room until a semblance of a broader identity, cohesion, and cooperation developed. Once the kiddies are let out of the room - e.g. the GoI's lightening of its Hindi agenda and embracing of free market economics - the kiddies now stick together and work, if that makes sense.

Let me know if I can clarify anything for you.