At least 5,000 people marched to the offices of Ireland's socially conservative prime minister on Saturday to call for clearer guidelines on abortion following the death of a woman denied a termination.

It was the largest of a wave of protests across Ireland in recent days in response to the death of 31-year old Indian woman Savita Halappanavar who died of septicaemia following a miscarriage 17 weeks into her pregnancy.

The Irish health authority (HSE) has launched an inquiry into the death, which has reopened a decades-long debate over whether the government should legislate to explicitly allow abortion when the health of a mother is at risk.

Activists in the overwhelmingly Catholic country, which has some of the world's most restrictive laws on abortion, say the refusal by doctors to terminate the pregnancy earlier may have contributed to Halappanavar's death.

"A vibrant, healthy woman starting her family life has died needlessly ... because of the failure of successive governments to deal with this issue," independent member of parliament Clare Daly told the crowd, which responded with chants of "shame."

Irish law does not specify exactly when the threat to the life or health of the mother is high enough to justify a termination, leaving doctors to decide. Critics say this means doctors' personal beliefs can play a role.

Despite a dramatic waning of the influence of the Catholic Church, which dominated politics in Ireland until the 1980s, successive governments have been loath to legislate on an issue they fear could alienate conservative voters.

Prime Minister Enda Kenny, whose ruling Fine Gael party made an election pledge not to introduce new laws allowing abortion, on Friday said he would not be rushed into a decision on the issue.

Halappanavar was admitted to hospital in severe pain on October 21 and asked for a termination after doctors told her the baby would not survive, according to her husband Praveen.

The foetus was surgically removed when its heartbeat stopped days later, but her family believes the delay contributed to the blood poisoning that killed Halappanavar on October 28.

"I just feel outrage," said Mary Sheehan, a midwife in her 50s, who took part in the march with a sign that read "Vatican Republic killed Savita. "I want the message to out her parents that the Irish people are demanding change."

The crowd also targeted the government's junior coalition partner, the Labour Party, which is more socially liberal, for not doing more to force change on the issue, chanting "shame on Labour."