The majority of India's citizens are Hindus, although with more than a billion people, India is a plural society with many, many religions. According to the World Factbook, about four in five people in India are Hindus, while just over 13% are Muslims and roughly 2% Christians and 2% Sikhs.
Just as in the US, however, much of the nation is secular, meaning they claim Hinduism as a religion, but do not follow it strictly. There is also considerable tension between Hindus and Muslims and Hindus and Sikhs, and that has led to rebel movements, violent riots and even civil war.
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To give a simple answer to your question, most people living in India are Hindu by religion. But, at the same time, you should remember that Hinduism is not an institutionalized religion, like Islam, Christianity & Buddhism. No one can be a Hindu except by birth. Hinduism is more of a faith than a systematized religion.
The name 'Hindu' has been derived from 'Sindhu' which is just another name of the river 'Indus', and the ancient civilisation of India is called the 'Indus Valley Civilisation'. During the last few years, some Hindu fundamentalists, like their Islamic counterparts, have been trying to revive the idea of Brahmanic Hindutva for their religio-political gains.