Savita Halappanavar died in hospital on 28 October last year from suspected septicaemia.
The 31-year-old's husband, Praveen, has claimed that doctors at Galway University Hospital refused to carry out an abortion 17 weeks into her pregnancy because a foetal heartbeat was present.
He alleged they were told Ireland "is a Catholic country".
The inquest into Mrs Halappanavar's death was opened before the Coroner for Galway city, Dr Ciaran MacLoughlin, on Friday morning.
He urged all sides involved in the hearing to respect the functions of the court after medical records stating that Mrs Halappanavar had requested an abortion were leaked on Thursday night.
The family's legal team had previously said that medical notes they had seen did not record the request for a termination.
Dr MacLoughlin offered his condolences to Mr Halappanavar and said he would carry out the inquiry with dignity and respect "his beloved Savita".
"It is my duty as coroner to ensure that the inquiry shall be independent, effective and prompt - that the procedures are open, transparent and accountable and are subject to public scrutiny," Dr MacLoughlin said.
The hearing, which is expected to last more than a week, will begin on 8 April at Galway Courthouse.
Meanwhile, two separate inquiries are continuing into Ms Halappanavar's death.
Independent health watchdog, the Health Information and Quality Authority (Hiqa), is examining "the safety, quality and standards of services provided by the HSE to patients, including pregnant women, at risk of clinical deterioration and as reflected in the care and treatment provided to Savita Halappanavar."
The findings of the probe, which is being led by four consultants from outside the Republic, will be made public when complete.
The Health Service Executive (HSE) is also carrying out an internal investigation and gave Irish Health Minister James Reilly an update before Christmas.
"The work of the investigation team is well advanced and it is anticipated that their report will be completed early in 2013," an HSE spokeswoman added.
However Mr Halappanavar, a 34-year-old engineer at Boston Scientific in Galway, has refused to co-operate with either probe and has threatened to take the Government to the European courts in pursuit of a public inquiry.
Elsewhere, the Irish Government has committed to legislate and introduce regulations to allow abortion if there is a real and substantial risk to a woman's life, including the threat of suicide.
Hearings took place before the Oireachtas Health Committee in Dublin last week.