The convicted friend is Hjalti Sigurjón Hauksson, who was found to be continually raping his stepdaughter on a daily basis since the age of five.
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.
In a short statement published on their website, the party - which holds only four seats in the 63-seat parliament - said it was withdrawing from the coalition over a "serious breach of trust within the government".
Prime Minister Bjarni Benediktsson took office in January with his Independence Party, the Reform Party and the centrists.
Benediktsson, who was implicated in the Panama Papers scandal that revealed offshore tax havens, had delayed the release of a report on tax evasion during the legislative campaign. The country suffered through years of economic upheaval after its debt-swollen banks collapsed during the 2008 financial crisis but now is experiencing a tremendous surge in tourism by those eager to see its pristine glaciers, fjords and waterfalls and the Northern Lights.
He said he would be looking to hold the election in November though that would mean it would not be possible to finish next year's budget.
The news knocked more than 1 per cent off the value of the Icelandic crown against the euro and the United States dollar. "The reason for the split is a serious breach of trust within the government".
"This situation was uncalled for but we will have an election", Benediktsson told a news conference in Reykjavik.
A letter of recommendation is among the requirements to apply for the process.
"I think if there's going to be another election and possibly the chance of a more left-wing government, then that probably decreases the chance of monetary policy being loosened any further in the future", Brown said.
Mr Benediktsson is reported to have been aware of his father's efforts since the summer, but the information has only now become public following a ruling from a parliamentary committee.
In his statement of apology, Mr Sveinsson tried to explain his actions, saying that Hauksson had brought him an already drafted letter which he signed.