The archbishop personally voted "No" arguing that gay rights should be respected "without changing the definition of marriage".
"I ask myself, most of these young people who voted yes are products of our Catholic school system for 12 years. I'm saying there's a big challenge there to see how we get across the message of the Church," he added.
Ireland is the first country in the world to legalise same-sex marriage through a popular vote, and its referendum was held 22 years after homosexual acts were decriminalised in the Republic of Ireland.
Among those voicing their approval of the outcome was UK Prime Minister David Cameron who tweeted: "Congratulations to the people of Ireland, after voting for same-sex marriage, making clear you are equal if you are straight or gay.