"I can see two solutions: a profound change to government or early elections," Kiska said in a televised tribute to slain reporter Jan Kuciak, who had been probing alleged high-level political corruption before he was found shot dead last week.
Kuciak's murder has raised fresh concern about media freedom and corruption both in Slovakia and Europe.
Kuciak's employer published his latest investigation this week, which appeared focused on fraud cases involving businessmen linked to Prime Minister Rober Fico's governing SMER-SD party.
Police have said his death was "most likely" related to his journalism work.
The government's response to the killing has drawn criticism in spite of several high-profile resignations.
Authorities this week detained several Italian businessmen named by Kuciak but on Saturday they were released due to lack of evidence.
Kiska, who is Slovakia's head of state, lamented what he termed the "arrogance of power" among senior officials and criticised the lack of action to restore public confidence in the government.
"I've waited a week to see what political measures the government would take to help ease tensions and re-establish trust," he said.
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