Halappanavar, a 31-year-old dentist, repeatedly asked staff at University Hospital Galway in western Ireland to terminate her pregnancy because she had severe back pain and was miscarrying, her family said.
But they replied she could not have an abortion because Ireland was a Catholic country and the foetus was still alive, her husband Praveen told the Irish Times.
"Savita was really in agony. She was very upset, but she accepted she was losing the baby," the 34-year-old told the newspaper by telephone from the Karnataka region of southern India.
"When the consultant came on the ward rounds on Monday morning Savita asked if they could not save the baby, could they induce to end the pregnancy.
"The consultant said, 'As long as there is a foetal heartbeat we can’t do anything.'
"Again on Tuesday morning, the ward rounds and the same discussion. The consultant said it was the law, that this is a Catholic country.
"Savita said: 'I am neither Irish nor Catholic' but they said there was nothing they could do."
Halappanavar died of septicaemia, or blood poisoning, on October 28, a week after she was admitted. The foetus had been removed on October 23 after its heartbeat stopped.
The hospital said it would begin a review of her death as soon as it could consult with her family, who are in India for her funeral.
Halappanavar was a Hindu originally from India. She and her husband lived in Galway, where he worked as an engineer.
"The fact that this young woman lost her life is a personal tragedy and a family tragedy. No words of ours here can deal with that loss," Kenny told parliament on Wednesday.
He said the health minister had asked for a report, while investigations had been launched by the hospital and by Ireland's Health Service Executive.
Kenny said he would not pre-judge the inquiries, but stressed: "It is very important and imperative that the standards that apply in our maternity units be kept at the very highest level of professionalism and competence."
Savita's death triggered numerous events around the country late Wednesday, including a protest outside parliament in Dublin.
One of the people in attendance at Leinster House, Helena Kelly -- who is the same age as Savita -- said she was appalled at hearing the news of the death.
"I would like to start my own family in a few years and to think things like this still happen absolutely terrifies me," she told AFP.
Meanwhile, hundreds of people took part in a candlelit vigil in Cork city.
People gathered outside the city's Opera House where dozens of candles spelling out Savita's name were laid on the ground.
Ireland's abortion laws have been the subject of debate for years.
Under a 1992 Supreme Court ruling, women in Ireland are legally entitled to an abortion when it is necessary to save the life of the mother. But legislation has never been passed to reflect this.
Health minister James Reilly has promised to introduce legislation during the term of this government, and is awaiting a report from an expert group.
The opposition Sinn Fein party urged the government to act quickly.
"I know there are strongly held views on the issue of medical termination, but the people spoke in referendums and firmly placed the onus on the (parliament) to deal with the issue by means of legislation," leader Gerry Adams said.
A 1982 referendum acknowledged the "right to life of the unborn... with due regard to the life of the mother," while a second in 1992 added an amendment that permitted the right to travel abroad for an abortion.
The European Court of Human Rights condemned Ireland in December 2010 for forcing a pregnant cancer sufferer, who feared that having a baby would worsen her health, to have an abortion abroad.