Mr Kenny described the report as 'another appalling litany of shame' and said apologies are not good enough.
He said this was a case where men protected guilty men, people in authority knew what was going on, and it was not confined to the 1950s but went on to the 1980s.
Mr Kenny said people who were in positions of authority and are still in those positions should not continue from any perspective of integrity and moral authority to lead.
Meanwhile, agencies providing support services for victims of abuse said calls to their helplines have increased after yesterday's publication of the Commission of Investigation into the Catholic Archdiocese of Dublin.
The Murphy Report found that hundreds of cases of abuse were covered up by the Archdiocese and other church and state authorities.
The report details a litany of rape and attacks on children by priests in the Dublin Diocese from 1975 to 2004. It revealed the systematic way that abuse was facilitated and covered up by the Archdiocese of Dublin and other church authorities.
Since the report was made public, agencies responding to calls for support have been very busy.
The Dublin Rape Crisis Centre has reported a three-fold increase in calls.
It said its helplines had been inundated with requests for help from people who had never before discussed the abuse they suffered
The Health Service Executive said around 50 victims contacted its National Counselling Service yesterday.
The other organisations providing help are Connect, One-in-Four, the Samaritans and Faoiseamh.
A small number of calls have been made to the dedicated garda information line set up following the publication of the Murphy Report.
Gardaí have asked that any person who wants to provide information in relation to the report or want to report information related to a criminal offence arising from the report to contact the Sexual Assault Unit at Harcourt Square in Dublin, on 01-6663066.
It is understood that so far fewer than 20 people have contacted the gardaí.
However they say they expect the number to increase as people read the report and come to terms with what has happened.
The Murphy report criticised some gardai for failing to properly investigate allegations of clerical abuse prior to 1995 but also found that the special investigation set up in 2002 was co-ordinated and comprehensive.
Forty files were sent to the Director of Public Prosecutions, 25 of those related to the Dublin Archdiocese.
Eleven priests and former priests were convicted of or pleaded guilty to sexual assaults on children and a number of others are still before the courts.
In the wake of the report, the Green Party said the Catholic Church must question its whole approach to sexuality.In a statement, Green Party TD Ciarán Cuffe says that at a time when members of the Catholic Church hierarchy are actively campaigning against the Civil Partnership Bill being brought before the Dáil next week, there must be a pause for reflection within the Catholic Church and a more concerted effort to protect the vulnerable.
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