Last month, the 53-year-old was ordained as Coadjutor Archbishop at Saint Patrick's Cathedral, Armagh and is set to succeed Cardinal Seán Brady, who is expected to retire in 2015.
Currently emerging from a long shadow of child abuse controversies, the Catholic Church faces many challenges at present - including the issue of abortion, falling numbers of worshippers and a distinct lack of young men and women taking up a vocation.
In an exclusive interview, Archbishop Martin told UTV that the Church cannot be complacent - that was one thing learned through the terrible, shameful, awful things that have happened in the past.
"And yes, it is true that it has damaged our, let's say, credibility in terms of being able to contribute to major debates today, and there is no doubt that I feel a lot of people say, 'Who are they to talk? Look what they were up to.'
He also said that church leaders are real people - and are not "squeaky clean".
"This idea that somehow you're going to find a church leader, myself or anyone else, who's going to be squeaky clean, who has no issues, I just don't think that happens because they're not real people."
The church is presently struggling to lead the way on the abortion debate, despite Cardinal Brady speaking out against it on numerous occasions.
Current law in the Republic of Ireland prevents abortion unless there is a risk to the life, rather than the health, of the mother.
In Dublin on Tuesday, the third and final day of Oireachtas Health Committee hearings on the Protection of Life During Pregnancy Bill 2013 ended.
The committee will now report to Minister for Health James Reilly before the Bill comes before the Dáil and the Seanad in June.
Archbishop Martin said: "The Church cannot keep from saying that abortion is wrong. For us, it is the deliberate and the intentional and direct taking of an innocent human life."
The next Catholic Primate added that public debate is impoverished if the voice of faith and the voice of people of faith is relegated away and not allowed to be heard.
"For me, and I think maybe the Troubles have had some influence on this, we tended to be quiet about religion. Don't mention it. I think that has maybe left a legacy for us, that we're afraid to talk about our faith."
One of 12 children - six girls and six boys - Eamon Martin was brought up in Pennyburn, Derry.
He said it was "a lovely thing" to grow up as part of a big family, before adding that his parents as "people of faith" were a very strong influence on him.
Former curate of St Eugene's cathedral in Derry, he was also principal of St Columb's College in the city for eight years.
At his ordination last month, the then Monsignor Martin made a call for renewal in the church and said he wants to reach out to those who have lost their faith.
"Of course there are some who will not want to listen," he said.
"There are others too who have been so hurt and betrayed in the past, that understandably they find themselves unable to trust our message."