The family of Savita Halappanavar say the Minister for Health, James Reilly, has until Thursday to agree to a public inquiry into her death or else they will take their case the European Court of Human Rights.
The father of the late Ms Halappanavar has made an appeal to the Government to consider a public inquiry.
Speaking from India, Mr Andanappa Yalagi said the family were not happy with the progress made so far. He added that he did not understand, or trust, the HSE investigation.
Dr Reilly has said that when he receives the reports of the two current investigations into her death, he will take whatever action is needed.
Earlier, Mr Reilly said he respects the views of Praveen Halappanavar and his right to do as he sees fit in seeking a different inquiry into his wife's death.
The minister said he has a duty of care to the women of Ireland and the west of Ireland to ensure practices at University Hospital Galway are safe.
He added that he has a duty to reassure them it is a safe place to have a baby and that he has to await the outcome of the internal and HIQA inquiries.
Minister Reilly said he will take whatever action those inquiries demand, but that "in fairness", this was the first maternal death at the Galway hospital in 17 years, and the service there has been safe.
Ms Halappanavar died on 28 October in UHG following a miscarriage.