The director-general of the Irish Prison Service, Brian Purcell, insisted overcrowding had not contributed to the death.
But opposition parties and prison officers warned the “overflowing” jail was descending into violence.
Douch’s body was found under a mattress yesterday morning in the cell he was placed in with six other “protected” inmates.
The suspected murder came weeks after the Prison Officers’ Association (POA) wrote to Mr McDowell highlighting the “dangerous, unhealthy and intolerable situation for staff and inmates” at the Dublin jail.
Mountjoy has witnessed a string of violent attacks and the POA directly blamed the minister’s closure of three other prisons for overcrowding at the facility.
Fine Gael’s Jim O’Keeffe demanded to know why Douch had not been isolated for his protection.
“The prisoner was put into a cell with at least six others, even though he had asked to be segregated for his own safety. This is the clearest example yet of how the prison system is buckling under the demand for more space,” he said.
Labour’s Brendan Howlin said it was an outrage that a prisoner seeking protection died in what was supposed to be a place of safety.
“Mr McDowell has closed other prisons and left Mountjoy overflowing; he needs to get a grip on what is happening in our prisons,” Mr Howlin added.
But Mr Purcell rejected the claims: “You are unfortunately going to get incidents like this and they cannot be put down to overcrowding. These things happen, within prison systems, throughout the world.”
Asked if he believed the prison service had failed in its duty of care to Douch, Mr Purcell said: “I don’t. Its difficult to get around that particular question when a prisoner has asked for protection and that prisoner now unfortunately is dead.
“We are going to have a close look at the systems we have in place because I suppose, to some extent, it’s difficult to avoid that conclusion.”
Mr McDowell said the inquiry would be conducted by Michael Mellett of the Independent Monitoring Commission.
“The death of Mr Douch, while in the care of the prison service, is a tragedy for his family and is greatly regretted by me.
“A full garda investigation is under way but the further enquiries by a former senior public servant should help identify any system failure which may have occurred,” the minister said.