Together the three parties held the slimmest of majorities 32 of the 63 seats in parliament following the October 29 election, which was called after the former prime minister resigned amid protests over his offshore holdings that were revealed in the Panama Papers leak.
Bright Future accused Benediktsson of failing to tell the government that his father had signed a letter supporting a convicted paedophile's bid to have his criminal record erased after serving five-and-a-half years in prison.
His father, an entrepreneur named Benedikt Sveinsson, had given support to a man who was convicted in 2004 of having raped his stepdaughter nearly every day for 12 years.
Calls for a snap election were made by most of the other parties in parliament, including the Pirate Party which also demanded a vote on constitutional changes and called the Independence Party "unfit for government".
The Icelandic crown fell roughly 1 percent against the euro and the dollar on Friday as a scandal involving the father of the country's prime minister left the government on the verge of collapse.
Benediktsson said the country should hold elections in November.
This old Icelandic system permits convicts to have certain civil rights restored - enabling them to run for public office, qualify for certain government jobs or serve as an attorney or solicitor, for example - if three letters of recommendation from persons of good character are provided.
Iceland's Prime Minister Bjarni Benediktsson speaks during a press conference in Reykjavik, Iceland September 15, 2017.
"The board of Bright Future has made a decision to terminate cooperation with the government of Bjarni Benediktsson", the party said in a statement.
All that is required for "restoring honour" is a letter of recommendation from a close friend.
The prime minister told reporters: "We have lost the majority and I don't see anything that indicates we can regain it".
The premier said on Friday that he was "shocked" to hear of his father's letter, never tried to hide anything and was never part of the decision process in the cases in question.
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.
"What was supposed to be a small gesture of good will towards a convicted criminal has instead turned into a continuation of the tragedy for his victim. For this I again apologise".