I'm not so sure that it is fully accurate to say India did so. At the time of independence, India also comprised of what are now Pakistan and Bangladesh. When British India gained independence in 1947, it was decided that the Islamic majority areas would become the state of Pakistan and the Hindu majority areas would become the state of India. Bangladesh became a part of Pakistan, not gaining independence until the 1970s after a bloody conflict.

Of course, Pakistan had many Hindus, and India had many Muslims. As a result, large scale ethnic migrations occurred as Hindus fled to India and Muslims fled to Pakistan. These migrations were accompanied by large scale mob violence, as people lashed out at what were now unpopular minorities and long simmering ethnic tensions exploded from the catalyst caused by cutting the country in two upon ethnic lines. The brand new states of India did not know exactly what to do with millions of refugees each, which led to issues of squalor and violence in impoverished refugee camps.

Then we have the issue of Kashmir. Jammu and Kashmir was a princely state, which basically meant it had a high level of autonomy, and the rulers of the princely states were allowed to choose whether to join Pakistan or India. Jammu and Kashmir was majority Muslim, so there was an expectation that it would join Pakistan, but the Hindu prince wanted to maintain independence. He made offers to maintain the status quo, which proved unacceptable to India, and lost part of his territory to a pro-Pakistani rebellion. As a result, he ended up accepting Indian rule. So far, there have been three wars and countless skirmishes between India and Pakistan over Kashmir, as well as insurgency against Indian forces, with India and Pakistan now aiming nukes at each other. The conflict seems to have cooled off in recent years, at least, but the underlying disputes remain.

So, I would not characterize India as having avoided war. Non-democratic rule yes, but a great deal of blood has been shed in the wake of the Partition of India. I lack the knowledge base to explain why India has remained a democracy, nor can I explain why India is more internally stable than Pakistan, but I can tell you that the Partition was not at all a peaceful event, nor have India and Pakistan coexisted in peace since partition. From the perspective of asking why India has avoided war compared to other former colonies, I would have to say that it didn't.