After resisting calls to resign and riding out several previous scandals, interior minister Robert Kalinak resigned on Monday, depriving Mr Fico of one of his closest allies and a co-founder of their populist Smer party.
His is the third resignation in a matter of weeks. They were the biggest protests in Slovakia since the 1989 anti-communist Velvet Revolution.
Kalinak's move comes after tens of thousands of protesters across Slovakia demanded the resignation of the government and a thorough investigation into the shooting deaths of Jan Kuciak and Martina Kusnirova.
Kuciak claimed that Italian businessmen with links to the Calabrian organised crime syndicate 'Ndrangheta had settled in eastern Slovakia, and had spent years embezzling European Union funds.
Kuciak had been looking into fraud cases involving politically-connected businessmen.
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The week before, Maria Troskova, the prime minister's chief state adviser, and Viliam Jasan, the secretary of the state security council, quit amid allegations about past dealings with the man. He was subsequently released and denies wrongdoing. Fico is leader of the leftist Smer-Social Democracy party.
"We won't dance on the graves of two young people, unlike opposition, media and, now, the president", he said.
The junior coalition partner Most-Hid group made Kalinak's resignation a central requirement for its continued support for the government. That party's leadership will meet on Monday to decide whether it will remain in the coalition.
Bugar said Fico rejected the Most-Hid party's proposal for a snap vote.
Kalinak said he hoped that by resigning, "I will contribute to the stabilization of the situation in Slovakia".