According to the Reykjavik Grapevine, the sex offender in question was Hjalti Sigurjón Hauksson, who was convicted of raping his stepdaughter nearly daily for 12 years.
Such letters are used in Iceland for "restoration of a honor" - a civil procedure, which allows people convicted of major crimes to regain employment opportunities.
Prime Minister Bjarni Benediktsson's Independence Party was left with only a one-seat majority after last October's election, and was forced to form a coalition with two other parties: the Bright Future party and the Independence Party.
It said in a Facebook post that there was "a serious breach of trust" behind its departure.
Sveinsson said he signed his name to the letter but didn't write it.
It came after reports that Benedikt Sveinsson, the father of Prime Minister Bjarni Benediktsson, had offered a letter of recommendation to "restore the honour" of a convicted paedophile. Iceland's economy is booming after the government earlier this year dismantled the last of the capital controls that had been in place since 2008.
Benediktsson is a former finance minister who was also named in the Panama Papers as having offshore accounts.
That leaves the country, whose economy was wrecked by the collapse of its banking system almost a decade ago, facing its second snap election in less than a year.
It is ultimately up to President Gudni Johannesson, whom Benediktsson will meet with on Saturday, to make the decision on a new election.
Friday's news knocked the Icelandic crown as much as 1 percent lower against the euro and the dollar, according to Reuters.
REYKJAVIK (Reuters) - Iceland's prime minister said on Saturday the president had accepted his request for a snap parliamentary election, and November 4 had been discussed as a possible date. "The reason for the split is a serious breach of trust within the government". Starting July, three cases of such recommendation letters granted to pedophiles have been disclosed, which have immediately triggered condemnation among the Icelandic society and calls for abolition of such practice.
When it emerged that some government officials had kept information from the public about a letter seeking to expunge Hauksson's record, the Bright Future Party said it was pulling out.