The centrist governing coalition is shepherding a bill through Parliament that would stop schools giving that sort of admissions preference, and would require religious schools to state publicly how they treat students who do not follow the school’s faith. The government also wants the church to relinquish control of some schools.

The archbishop of Dublin, Diarmuid Martin, warning that schools are gradually losing their Catholic character, has pushed for years for the church to divest itself of hundreds of schools. But he has had little success, meeting strong resistance from the religious orders that control them.

It is a measure of how much Ireland has changed that the loudest objections to the new policy on religion classes in state-run schools have come from the schools themselves. Teachers and administrators have objected not to having less religious instruction, but to the statement by the education minister, Richard Bruton, that the government will not provide funding for alternative classes.

“Ireland has changed tremendously, and in urban areas there will be significant numbers of people choosing other subjects rather than religion,” said Michael Moriarty, the general secretary of Education and Training Boards Ireland, the organization created by the government that oversees most state-run schools. “We don’t have the staffing resources to put on extra classes with extra teachers.”

The teachers’ union went a step further, not only calling for more money and more teachers to carry out the policy, but also asking why “such a measure would only be implemented for a proportion of the country’s second-level schools” and not the religious ones.

For generations, Ireland was deeply conservative and its governments ceded much of the public service sector — schools, hospitals and orphanages, among other institutions — to the church. In recent years, the church’s role and reputation have been eroded by revelations about how its people and institutions abused that power — including widespread sexual abuse and the mistreatment of unwed mothers and their children in homes that resembled prisons.

At the same time, the country has become much more liberal on social issues, often in ways opposed by the church. In the 1990s, Ireland decriminalized same-sex relations and legalized divorce.

It legalized same-sex marriage by referendum in 2015, with 62 percent of voters approving the change. And in May, the country will vote on a referendum to repeal the constitutional ban on abortion.