The abortion issue is one of the most contentious issues facing the Government, with Labour and Fine Gael TD split down the middle.
Former Junior Health Minister Roisin Shortall, who resigned in September, has called for an independent inquiry into the tragedy.
"I think there is a need for the Government to appoint somebody outside the hospital to investigate this case," she told the Herald.
"This is an appalling tragedy that has arisen as a result of a lack of clarity on what hospital staff can and cannot do in these situations.
"Essentially government legislation has been avoided for over 20 years, and I hope this Government is the one that will bring clarity and legislation and follow the advice of the expert group," she said.
A spokesman for the Department of Health said the department and its ministers wished to extend their sympathies to the family of Mrs Halappanavar.
He said there were currently two investigations into what had taken place at the hospital in Galway and the health minister and the department would await the outcome of those investigations before making a statement.
A protest is set to be staged outside the Dail this evening as the campaign mounts for legislation to reflect the 1992 ruling of the X Case. The Action On X group is critical of the Government role in delaying on legislation to deal with abortion where the life of the mother is at risk.
"It's absolutely outrageous that 20 years after the X Case, successive governments have failed to legislate in this area," said spokesperson Sinead Kennedy. "Some of the responsibility for this lies with the Fine Gael and Labour Government," she added.
The United Left Alliance said that responsibility for the death of Savita Halappanavar lies squarely with Labour and Fine Gael.
"This event is horrendous and will shock people the length and breadth of the country," said Socialist Party TD Clare Daly. "This is an outrage which demands immediate action.
"Sadly, the very thing we feared last April when we put our X Case Bill before the Dail, has happened. A woman has died because Galway University Hospital refused to perform an abortion needed to prevent serious risk to her life," she said. "This is a situation we were told would never arise."
A medical adviser to the Pro-Life movement, Dr Berry Kiely, said: "I would urge caution that the circumstances of this tragedy are not whipped up into a controversy concerning abortion. The medical treatment for a miscarriage is not an issue of controversy regarding abortion.
"I would not wish to comment further as I am not familiar with the details of this tragedy."