Two episodes show how directly religion has intersected with Slovak politics, one historical and one recent.
Jozef Tiso was a Catholic priest who became president of the Slovak State, the client regime that existed under German protection from 1939 to 1945. Under his government the great majority of Slovak Jews were deported, and the state paid Germany a per capita sum for taking them. Tiso was tried and executed in 1947. The Vatican had protested the deportations during the war, and the Slovak bishops issued a pastoral letter in 1943 against them, but Tiso himself did not resign and continued in office. The Catholic Church in Slovakia has addressed the period, including a formal apology for the church's share of responsibility, and yet commemorations of Tiso by a nationalist fringe continue and are periodically the subject of prosecutions. The argument is not really about the facts of the deportations, which are settled, but about whether a priest who led that state can be claimed as a Slovak patriot.
The second episode is the conscience clause. The 2000 treaty with the Holy See provided for a further agreement on conscientious objection, and a draft was prepared that would have allowed institutions and individuals, including in the health system, to refuse participation in procedures contrary to Catholic teaching. Legal opinions commissioned by the European Union raised concerns about its breadth. When the Christian Democratic Movement pressed it in 2006, the prime minister refused, the party left the coalition, and the government fell, forcing an early election.
A treaty commitment brought down a government. That is an unusually direct measure of how much these arrangements weigh.