Bright Future accused Prime Minister Bjarni Benediktsson of failing to inform the government that his father had signed a letter supporting a convicted paedophile's bid to have his honour restored.
Benediktsson's father, an entrepreneur named Benedikt Sveinsson, backed a man who was convicted in 2004 of having raped his stepdaughter almost every day for 12 years so he could erase his criminal record.
Iceland could now face a second snap election in a year.
The last government collapsed over the Panama Papers scandal that embroiled several ministers and forced former prime minister Sigmundur David Gunnlaugsson to resign.
The ruling centre-right coalition of the conservative Independence Party, led by Benediktsson, the centre-right Reform Party and the centrist Bright Future, was formed only nine months ago after the October vote.
Bright Future said its executive committee "has decided to terminate the cooperation with the government" because of a "serious breach of trust".
Reform Party leader Benedikt Johannesson meanwhile called for new elections.
Icelandic media speculated the minority government could stay if it gains the support of another party, or President Gudni Johannesson could ask another party to form a coalition, or new elections could be held.
"The prime minister will most likely resign today," Olafur Hardarson, a political science professor at the University of Iceland, told AFP.
"Usually, the president would then ask the sitting government to continue as a temporary government," he added.
After serving his sentence, the convicted child molester had applied for "restored honour" to enable him to expunge his criminal record, which is permitted under Icelandic law and is subject to the support of persons of good character.
According to media reports, the prime minister's father is an old friend of the man, who served a five-and-a-half year jail term.
Smari McCarthy, a lawmaker for the anti-establishment Pirates Party, said: "Iceland's Jimmy Savile case: our PM, who was in the Panama Papers, has hid for two months his father's support for a paedophile's clemency," referring to British DJ and television presenter Jimmy Savile who abused scores of girls and young women.
Iceland's justice minister reportedly informed the prime minister about his father's support for the molester in July and he had kept it hidden until a parliamentary committee brought the case forward.
"This was the last straw,"Gudlaug Kristjansdottir, Bright Future chairwoman, was quoted as saying by the Iceland Monitor daily.
Benediktsson, who was implicated in the Panama Papers scandal that revealed offshore tax havens, has delayed the release of a report on tax evasion during the legislative campaign. His coalition partners now feel the trust has been broken.
'Knocked on doors for favours'
The Icelandic procedure for rehabilitating convicts, which does not change or reverse a verdict, is at the heart of a fierce public debate that has intensified in recent weeks over another convicted paedophile who was granted an expunged record.
More and more Icelanders see this procedure as a blow for victims.
The Icelandic Women's Rights Association said the government collapsed because women were willing to make their voices heard.
"People spoke out about violence which women and children were subjected to. People spoke out when convicted abusers knocked on their friends' doors to ask for favours," it said in a statement.
A spokeswoman of the Pirate Party on Friday called for an urgent vote for constitutional reforms, without specifying the changes it sought.
"The Pirates Party calls on all other parliamentary parties to comply with this call and invites the president of Iceland ... to approve a new constitution before the parliamentary assembly is suspended," Birgitta Jonsdottir wrote on Facebook.