Archbishop of Armagh and Primate of All Ireland Eamon Martin also said Catholic schools could be found in breach of the Constitution if they continued teaching that marriage is only for men and women.
He said the Church was still undecided about whether priests would cease performing the civil registration of marriages at Church weddings but said that would also become an issue if there was a yes vote.
Archbishop Martin said that, contrary to what voters were being advised, a yes vote would not merely expand the definition of marriage but “radically change” it.
He acknowledged that marriage was not defined in the Constitution, but said it was clearly understood when written that marriage was between a man and a woman.
He said that, regardless of the outcome of the referendum, the Church would continue to teach this and he said this could bring some of the services it provides — such as teaching and counselling — into conflict with the law.
“We provide a very successful and very helpful marriage counselling service and marriage preparation service but we do so for men and women,” said Archbishop Martin.
“If we are asked to provide it for men and men or women and women in preparation for marriage, we will have to say no. That’s not within our sincerely held beliefs.
“We are receiving some State funding and there is a risk to that funding would we be seen as discriminatory.”
He said that, in the UK, the Church had to withdraw from providing adoption services because it would not place children with same-sex couples.
Archbishop Martin was speaking on RTÉ radio after releasing the Church’s first comprehensive statement on the upcoming referendum in which he criticised politicians for failing to articulate the concerns of the no side.
“How is it that many people won’t even raise these issues in their families and workplaces for fear of being ridiculed or condemned as homophobic? Could we not expect at least some of our legislators to engage in public discussion on both sides of this debate?” he wrote.
The Yes Equality campaign invited the Archbishop to a public debate on the issues, claiming he had confused the distinction between civil marriage and church marriage.
Spokeswoman Gráinne Healy said: “In England, Wales, and Scotland, the Catholic Church have continued to perform civil marriage as part of the religious service following the introduction of civil marriage equality there.”
Communications Minister Alex White also rejected Archbishop Martin’s claims, saying there would be no restriction on church teachings in schools.
“That of course was the case after we changed the Constitution to introduce divorce,” said Mr White. “It’s still something that the Catholic Church are opposed to. And they are still free to teach that in schools.”
Tánaiste Joan Burton said while the Church facilitated the civil registration of marriages, they were not responsible for it: “Whatever the Catholic Church wishes to do in their wisdom is a matter for the Catholic Church.”