|Ms Caruana was killed when a car bomb exploded near her home in Malta|
Ms Caruana Galizia - who reported extensively on corruption on the island and elsewhere, including through the Panama Papers leak - was killed in a car bomb near her home in Malta on Monday.
Her sons Matthew, Andrew and Paul Galizia dismissed a government's offer of a reward to help find their mother's killers - and instead attacked Prime Minister Joseph Muscat.
Writing on Facebook, they said the Prime Minister had presided over the "birth of a society dominated by fear, mistrust, crime and corruption" and said authorities could not guarantee a just investigation into the killing.
Ms Caruana Galizia was facing lawsuits as a result of her work
"The Prime Minister asked for our endorsement. This is how he can get it: show political responsibility and resign," they wrote.
"Resign for working to cripple our mother financially and dehumanise her so brutally and effectively that she no longer felt safe walking down the street."
An investigative journalist renowned for exposing corruption among the political elite, Ms Galizia had had her bank accounts frozen and faced libel suits from those in government as a result of her work.
Her death was mourned with vigils attended by thousands of Maltese, and authorities have vowed to carry out a full investigation, while expressing horror at the killing.
A candlelight vigil held to protest against the assassination
Roberta Metsola, an MEP for the opposition Maltese People's Party, told Sky News that the Prime Minister bore responsibility for the killing.
"The Prime Minister has tried to come across as being a victim of Daphne's criticism but the reality is she has levelled extremely grave accusations against him and his government," she said.
"At the moment there is fear in our country and there is great scepticism about whether the Prime Minister and his government are genuinely trying to seek justice."
Writing on Facebook, the family said the institutions their mother criticised would be unlikely to carry out a just investigation.
"Justice, beyond criminal liability, will only be served when everything that our mother fought for - political accountability, integrity in public life and an open and free society - replaces the desperate situation we are in," the post said.
Mr Muscat has ruled out quitting and has vowed to bring those responsible for the killing to justice, with the help of FBI investigators.
On Wednesday he told parliament that the government would put up a "substantial and unprecedented reward," for information leading to a conviction over the killing.