Slovakia has seen the largest protest since 1989, with demonstrators calling for the the government to step down.
However, Mr Fico's junior coalition partner, the Most-Hid group, demanded the resignation of Mr Kalinak in exchange for continued support.
On Tuesday, the Slovak prosecutor general's office confirmed that it was investigating Mr Kalinak over allegations from a former prosecutor that the erstwhile minister and top police officials had sabotaged a major corruption inquiry.
Kalinak said he hoped that by resigning, "I will contribute to the stabilization of the situation in Slovakia". "For this reason I have made a decision to resign as deputy prime minister and interior minister". Meanwhile, Prime Minister Robert Fico opposes the early elections. Authorities estimated 50,000 people had rallied in Bratislava out of a population of 5.4 million. Police said the deaths were "most likely" related to Kuciak's unfinished investigation linking senior Slovak politicians to the Italian mafia.
Bela Bugar, chairman of the Most-Hid party of mostly ethnic Hungarians, said that if its two governing partners don't agree to negotiate an early vote, his party is ready to leave the coalition.
Kalinak is Fico's key ally in Smer-Social Democracy, which has previously been alleged to have links to corruption.
Kalinak's move to quit is one of Fico's biggest political sacrifices of his three terms in power as he is also losing the man who was best placed to replace him at the head of his Smer party.
Critics argue Mr Kalinak, who oversees the police as interior minister, can not guarantee an independent investigation into the killing of Mr Kuciak and his fiancee. "We've found a deeply divided country that is almost traumatized", said Ingeborg Grässle, a German politician who serves as a member of the European Parliament.