He was responding to comments made by the Archbishop of Durban, Cardinal Wilfrid Fox Napier, who said in a BBC interview that paedophilia was an illness, not a crime.
The South African prelate also questioned whether someone with this psychological defect deserved to be automatically punished.
However, Dr Martin criticised Cardinal Napier's view, saying the fact that someone who has abused a child might have psychological disturbances did not exempt them from the rigours of the law.
Harm
"No one has the right to harm a child. And no one can excuse themselves from that," he told the Irish Independent.
He was speaking at Dublin's Pro Cathedral after a Mass in thanksgiving for the election of Pope Francis, which he concelebrated with the papal nuncio, Archbishop Charles Brown.
The nuncio said he was "absolutely certain" that Pope Francis would continue to take a very strong line on child sexual abuse.
"I think all the signs are that Pope Francis will be a courageous reformer in the church," he added.
Separately, Dr Martin moved to scotch accusations that the newly elected Pope collaborated with Argentina's military junta (1976-1983).
"All of those matters have been investigated. I'm sure that nobody can ever say that they did enough in the circumstances but to incriminate or say that this man acted wrongly – the evidence is pointing in another direction," he said.
Asked if the pontificate of Pope Francis would see the appointment of women to more prominent roles in the Catholic Church, Dr Martin responded: "I think there is absolutely no doubt. The papacy will give a lead, will set a tone, but much of this work has to be done in the local churches and this is where the church in Ireland has to question itself and look forward."
However, in his homily, Dr Martin warned against thinking the new Pope had come to support any particular line in church politics, but added everyone had been struck by the humility and simplicity of Pope Francis and his constant stress on personal poverty, love for the poor and a church characterised by poverty.
He also addressed the Irish church's long history of a "very strong judgmental trait".
Church
leaders, individuals and communities of all sorts, often thought that
their own judgmentalism was justified, he acknowledged.
He
suggested that today's liberals were also guilty of a form of reverse
judgmentalism which was equally willing to excommunicate others who
thought differently.