Patient Focus is calling for an independent inquiry into the deaths of four newborn babies at the Midland Regional Hospital in Portlaoise.
An RTÉ 'Prime Time' documentary, broadcast last night, investigated the deaths which occurred over a six-year period.
It revealed the infants died shortly after birth, although they were not suffering from any abnormalities at the onset of labour.
The HSE has apologised for shortcomings in the cases and the National Director of Acute Hospitals, Ian Carter, will arrange to meet the families.
Minister for Health James Reilly has requested a full report.
Patient Focus said that an independent investigation is needed, going back as far as 10 years and examines care in all maternity units.
Spokesperson Bridget Doherty said that the apology from the HSE is not enough.
"They need to work out why they did not contact the families, why families weren't informed of possible failings, and that reviews were taking place," she said.
"Anybody who is damaged by the healthcare system - you can't get the full story if you don't involve who is being damaged, or possibly damaged."
Reform Alliance TD Lucinda Creighton has called for a HIQA inquiry into the deaths.
"HIQA has proven itself to be an independent investigator, to be, I suppose, removed, at arm's length from the HSE and from the hospitals concerned," she said.
"So I think HIQA is the appropriate body, which can demonstrate its independence and ultimately find out precisely what happened and ensure that recommendations are made."