Light tributes placed during a silent protest in memory of murdered journalist Jan Kuciak and his fiancée, Martina Kusnirova. (Bundas Engler/AP)

HIGH-LEVEL CORRUPTION has been a scourge for years in Slovakia, the small Central European republic that is a member of NATO and the European Union. Stories of graft and shady dealings have earned political parties of all stripes the contempt of Slovaks, but the misdeeds hadn’t spilled over into bloodshed — until now.

Last month a young Slovak journalist, Jan Kuciak, was gunned down along with his fiancée in the apartment they shared, in what looked like a professional hit. Mr. Kuciak had been investigating ties between the Italian mafia and top officials in Prime Minister Robert Fico’s government. Mr. Kuciak’s murder has outraged Slovaks, who have taken to the streets to demand justice.

Whether Mr. Fico’s government can deliver it, or even survive, is a question that he himself deepened by delving into unsubstantiated conspiracy theories involving a rival politician who happens to be Slovakia’s president. Not exactly a promising start to a criminal investigation.

The prime minister’s jitters may have been exacerbated by the focus of Mr. Kuciak’s inquiries, which were published posthumously: how a onetime Miss Universe contestant and topless model came to occupy the post of chief state adviser in Mr. Fico’s government. The woman, Maria Troskova, was formerly a business partner of a suspected Italian mob figure living in Slovakia, and also an aide to a member of Mr. Fico’s governing party, Viliam Jasan, secretary of the country’s national security council. Ms. Troskova and Mr. Jasan resigned following the murders.

Mr. Kuciak, 27, wrote that Italian businessmen linked to the ’Ndrangheta, a powerful Calabrian organized crime syndicate, had infiltrated poor eastern areas of Slovakia, embezzling E.U. funds and laundering money while seeking government connections.

Mr. Fico took to television after the murders, bizarrely appearing with a pile of cash amounting to 1 million euros ($1.24 million), the reward for information leading to a conviction. That did nothing to settle speculation about what the culture minister, Marek Madaric (who also resigned, to protest the killings), called “a possible breach” of organized crime at “the level of state institutions.”

The combative Mr. Fico has a track record of belittling rivals and critics — he has called journalists “anti-Slovak prostitutes” — and spared no effort in making wild accusations following Mr. Kuciak’s death. In a televised statement, he accused Slovakian President Andrej Kiska, who called for a government shake-up following the slayings, of trying to destabilize the country, suggesting he was doing so in league with the American philanthropist George Soros. Mr. Kiska, who beat Mr. Fico in the 2014 presidential election, responded more plausibly by saying the prime minister had resorted to conspiracy theories to divert attention from the crisis.

The FBI and Britain’s Scotland Yard are helping Slovak authorities with the investigation. Last week, thousands marched through the Slovak capital, Bratislava, carrying photos of the dead couple as well as a banner proclaiming, “An attack on journalists is an attack on all of us.”