The 59-year-old Mr Fico took office in October after his centrist populist Smer party won a general election.
He is serving his fourth term as prime minister after campaigning on proposals for peace between Russia and Slovakia’s neighbour Ukraine, and for halting military aid to Kyiv, which his government later did.
Mr Kalinak said the government would carry on without Mr Fico “according to the programme he has outlined”, including two meetings next week.
The assassination attempt has deeply shocked the EU and Nato member country of 5.4 million people, already sharply divided over politics for years.
Outgoing pro-Western President Zuzana Caputova and her successor Peter Pellegrini, a Fico ally who will take office in June, have called on fellow Slovaks to refrain from “confrontation” after the shooting.
They called a meeting of all parliamentary party leaders for Tuesday in a bid to show unity in the aftermath of the attack.
Mr Kalinak, however, suggested Smer would snub the meeting.
“They invited political party chiefs and our chairman is in the hands of doctors,” he said.
Mr Kalinak added he would call Ms Caputova about the matter, stressing that Slovakia needed “reconciliation and peace”.
But he was among politicians pointing fingers at their opponents for the attack, slamming the opposition and “selected media” on Friday for labelling Mr Fico as a criminal, dictator or a servant of Russian president Vladimir Putin before the assassination attempt.
“All these lies are the main reason why Robert Fico is fighting for his life today,” he said in an emotional message on Smer’s website.
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