Italian prosecutors have begun an inquiry into the death after a miscarriage of a woman of 32 who was pregnant with twins.
The family of Valentina Milluzzo said the doctor treating her refused to abort the foetuses because he was a "conscientious objector" to abortion.
The hospital involved has categorically rejected the family's claims.
The woman's burial has been postponed while further investigations are carried out.
What went wrong?
Milluzzo was admitted to Cannizzaro hospital in the Sicilian city of Catania on 29 September after suffering complications and going into premature labour in her 19th week of pregnancy. She had had fertility treatment at another health centre.
She was in a stable condition in hospital for more than a fortnight but on 15 October her blood pressure and temperature dropped and her condition worsened.
According to the family's lawyer, one of the foetuses was suffering breathing problems. The lawyer alleges that the gynaecologist refused to abort the foetuses in order to save the mother and said: "As long as it's alive, I will not intervene."
No action was taken while the troubled foetus was still alive and hours later both had died, the lawyer said.
Overnight, the mother's condition deteriorated and she contracted an infection. On 16 October she was transferred to intensive care, where she died.
What does the hospital say?
Angelo Pellicano, the head of the Cannizzaro hospital, said the lawyer was wrong.
"There was no conscientious objection on behalf of the doctor that intervened in this case because there was no voluntary termination of the pregnancy, but (the miscarriage) was forced by the grave circumstances," he told the Ansa news agency. "I rule out that a doctor could have told the family what they say he told them."
Sources told Ansa that the doctor had not been registered as a "conscientious objector".
Catania's prosecutor has halted Milluzzo's burial while a post mortem examination is carried out. There are unconfirmed reports that some of the hospital's medical staff may be investigated.
A senior doctor at the hospital, Paolo Scollo, told the Corriere website that all the doctors in his department were "objectors", and external doctors were called in when necessary. "However, in this case we're talking about a spontaneous miscarriage, no external help was needed. So we do not think the doctor was negligent," he said.
Can a doctor be a conscientious objector?
It emerged earlier this year that the proportion of Italian gynaecologists refusing to carry out abortions in 2013 was 70%, according to Italian government figures. Italy is a predominantly Catholic country.
But in southern Italy the proportion was even higher and in Sicily it was 87.6%.
Abortion has been legal in Italy since 1978, provided it is during the first 12 weeks of pregnancy.
After 12 weeks it is allowed only if the life of the mother is at risk or there is a problem with a foetus.
If there is a chance the foetus can survive outside the womb, termination is lawful only if pregnancy or delivery is a serious risk to the woman's health.
More on abortion in Europe