Though India seemed to be as divided as Europe by the time the English colonized it, why has India operated as a united state without massive calls of secession or civil war

There's a problem with the premise.

To start it did break up. Pakistan split off in 1947 as East Pakistan – now Bangladesh – and West Pakistan – now Pakistan. These were formerly part of India until then. In 1971 Bangladesh then split from that.

Before that, Burma and Sri Lanka (then Ceylon) became separate in 1948.

But the other aspect, what I'm going to give an overview of here, is what you've just not heard about:

without massive calls of secession or civil war

Having gone on more or less continuously since since the mid-1960s, there's a significant insurgency that in the northeast. Even today you can't move freely through the region without having to go through a good number of military checkpoints and a heavy military presence through the Assam Rifles, a well established paramilitary force specifically developed to fight insurgency from the tribal groups during British rule.

Really, the Northeast has been dealing with separatists movements and calls for secession for much of Modern India's existence.

In Manipur, the United National Liberation Front was founded in 1964 advocating for a sovereign nation for Manipuris. The People's Revolutionary Party of Kangleipak was founded in 1977, and a year later, the People's Liberation Army (PLA) who I'll mention again later. Then 1980 saw the founding of the Kangleipak Communist Party. These are not all separatists in the same degree, but taken together with the UNLF they're part of a larger history of Manipuri insurgency and separatism.

In Assam there is the United Liberation Front of Assam, active since the 1970s. They're less active today, but have been in the news since one of their leaders was recently released in order to participate in talks with the central government. They've called for outright independence, but have mostly had to give into a lack of support among the Assamese people.

You also had/have a number of smaller ethnic separatists movements. The there is the National Democratic Front of Bodoland advocating for an independent state for the Bodo people. The NDFB got its start in 1986, but is much less well known than the ULFA.

There's also the Karbi Longri North Cachar Hills Liberation Front advocating for Karbi self rule, but they're only a decade old so moving on.

1989 was also the founding of the National Liberation Front of Tripura, a separatists group fighting for Tripuri independence. Tipura you may not have heard of, since it's one of the smallest states in the country and is almost entirely surrounded by Bangladesh.

You also have the Mizo National Front in Mizoram which in 1966 was behind the Mizo National Front uprising when they declared Mizoram an independent state and India responded with military force. The history of Mizoram is incredibly interesting and unfortunately there is much more going on in the 1960s-1980s than I can possibly speak to, but if you're interested in separatist movements, I strongly encourage looking into the Mizos. The tensions between the Indians and the Mizos has been significant, and has had long lasting repercussions, many of them linguistic. Even today you are not to speak Hindi in Mizoram, and Mizo's growing up there are essentially forbidden to learn it.

At the end of the conflict between India and the Mizos in 1986 and the founding of the State of Mizoram in 1987, you get the Hmar People's Convention, a group fighting for an autonomous Hmar region in Mizoram. They led violent insurgency from 1987 to 1992 in Mizoram.

But one of the biggest and very much still active groups is the Naga separatists. In the late 1940s and on, the Naga National Council (NNC) was engaged in violent insurgency, which lasted well into the 1970s. This began in 1947 when the NNC declared an independent Naga nation, which as we've already seen doesn't go over well with the Indian government. This eventually ended, for the most part, in 1975 with the signing of the Shillong Accord which ended the worst of the fighting.

Not everyone was pleased with having a state of Nagaland, however. Shortly after, the National Socialist Council of Nagaland (NSCN) became active to carry on what the NNC has originally been fighting for: A separate Naga nation. The NSCN was founded in 1980 based on dissatisfaction with the peace agreements and the concessions India had been willing to make to the Naga nationalist movement. Today there are actually three NSCN groups. There is NSCN-K which is led by S S Khaplang, NSCN-IM led by Isak Chishi Swu and Thuingaleng Muivah (and a new one just this past year, NSCN-R, the R being "Reformation"). A ceasefire was signed in 1997 with NSCN-IM. NSCN-K is a whole other matter but also far outside the 20 year rule.

Also going on in this period, China was heavily involved in bringing Naga separatists into China to provide arms and military training. They also brought members of the Manipuri People's Liberation Army, mentioned above, as well as other separatist/insurgent group members. The border is porous in that area, if you're accustomed to mountainous jungle terrain. This trainig began in 1967 but Chinese support for separatists in the northeast began more formally in 1962. Training of militants lasted all the way through the 1970s. Many of those arms would have been surrendered after the Shillong Accord, but some remained, and more home-made sorts of matchlocks were/are quite common. Note that gun ownership is formally restricted.

It's important to point out that these are not simply militants. They also operate civil governments, issue laws and edicts to the villages over a wide range of territory and collect taxes. This has typically put the villages in their area under extra pressure, much in the way that paying ransom to a terrorist organisation can legally be taken as financial support of terrorism. Breaking the 20 year rule I could tell you about more recent events related to these points. But the point is, they've attempted to develop a government, a sort of "Socialism with Baptist-fundamentalist characteristics", to borrow China's phrasing. They've operated as if their independence is already granted.

And of course there are many more such groups not mentioned here.

So as you see, there very much have been massive calls of secession and, depending on how you want to define it, civil war.

See also:

Misra, Udayon (2000) Periphery Strikes: Back Challenges To the Nation-State in Assam and Nagaland. Indian Institute of Advanced Study
Srikanth, H; Thomas, CJ (2005) Naga Resistance Movement and the Peace Process in Northeast India. Peace and Democracy in South Asia, vol. 1, no. 2.

edit: typos