A woman who died after being gang raped and beaten on a bus in India's capital has been cremated, amid an outpouring of anger and grief by millions across the country demanding greater protection for women from sexual violence.

Sunday's cremation took place during a private ceremony in Delhi soon after the woman's body arrived on a chartered Air India flight from Singapore, where she died at a hospital on Saturday after being sent for medical treatment.

After the body arrived at the airport, it was taken to the woman's home for religious rituals before being escorted by police to the crematorium. Security was tight, with no public or media access permitted.

Sheila Dikshit, the chief minister of New Delhi, and the junior home minister, RPN Singh, placed wreaths beside the body before it was cremated, the Press Trust of India news agency reported.

The prime minister, Manmohan Singh, and Sonia Gandhi, head of the ruling Congress party, were at the airport to receive the body and meet family who had also arrived on the flight.

Hours after the victim died early on Saturday, Indian police charged six men who had been arrested in connection with the attack with murder, adding to accusations that they beat and gang raped the woman.

Delhi police spokesman Rajan Bhagat said the six suspects face the death penalty if convicted. The case has triggered protests across India and raised questions about lax attitudes by police toward sexual crimes.

After 10 days at a hospital in New Delhi, the victim, who has not been identified, was taken on Thursday to Singapore's Mount Elizabeth hospital, which specialises in multi-organ transplants. She arrived in extremely critical condition, which then deteriorated. She died with her family and officials from the Indian embassy by her side, according to the chief executive of the hospital, Dr Kevin Loh.

Following her death, thousands of Indians lit candles, held prayer meetings and marched through various cities and towns, including Delhi, Mumbai, Bengaluru and Kolkata, on Saturday night to express their grief and demand stronger protection for women and the death penalty for rape, which is now punishable by a maximum of life imprisonment.

Singh said on Saturday that he was aware of the emotions the attack had stirred, adding that it was up to all Indians to ensure that the young woman's death will not have been in vain.

On the night of the attack, the woman and a male friend, who also has not been identified, were on a bus after watching a film when they were attacked by six men who raped her. The men beat the couple and inserted an iron rod into the woman's body, resulting in severe organ damage. Both were then stripped and thrown off the bus, according to police.

Gandhi, the ruling party chief, assured the protesters in a statement that the rape victim's death "deepens our determination to battle the pervasive, the shameful social attitudes and mindset that allow men to rape and molest women and girls with such an impunity".

Meenakshi Ganguly, South Asia director of Human Rights Watch, said the woman's death was a sobering reminder of the widespread sexual violence in India. "The outrage now should lead to law reform that criminalises all forms of sexual assault, strengthens mechanisms for implementation and accountability, so that the victims are not blamed and humiliated."

Singh said he understood the angry reaction to the attack and that he hoped all Indians would work together to make appropriate changes. "It would be a true homage to her memory if we are able to channel these emotions and energies into a constructive course of action," he said.