Most-Hid, a junior party in his coalition, raised the prospect of withdrawing support as it entered internal talks on Friday, after calling for Interior Minister Robert Kalinak, a close ally of Fico, whose business acquaintances had been subjects of Kuciak's earlier articles, to resign. The two were found shot dead in their home last weekend, representing the first murder of a journalist in the country.
His last, unfinished article was published posthumously by Slovak and global media.
Prime Minister Robert Fico is known for numerous attacks on the media.
Kuciak's death has swelled public anger about allegations of widespread corruption in the European Union member state and rattled Fico's three-party government.
The rallies are taking place amid calls for worldwide experts to join the investigation and for Interior Minister Robert Kalinak, who was linked to corruption scandals in the past, to resign.
"There is a dirty mess all around here. and we common people have to put up with it", protester Jozef Belovic, 58, said as he walked through the capital on Friday in sub-zero temperatures along with an estimated 10,000 to 15,000 others.
Troskova and the secretary of the country's Security Council, also mentioned in Kuciak's reporting, resigned on February 28 pending results of the investigation.
Demonstrators held photos of the 27-year-old journalist and carried banners with slogans such as: "Mafia get out of my country", and "an attack on journalists = an attack on us all".
Marches were held in 25 towns and cities across the country and overseas, including in London, Paris and Brussels.
Slovak authorities have reportedly released at least six of the seven people detained in connection with the murder.