Nicola Sturgeon said she felt a "huge sadness" about him quitting, saying he had been her friend and mentor for 30 years.
But she added that "the hard fact remains that two complaints were received by the Scottish Government that could not be ignored or swept under the carpet."
The Scottish Government has passed details of the complaints to Police Scotland, and the force has said it is assessing the information.
The Daily Record newspaper says it has seen the wording of one of the complaints, which was made by a Scottish Government staff member.
She alleges the former first minister touched her bottom and breasts through her clothing while she was alone with him at Bute House in December 2013, the paper reported.
Bute House is a listed building in Edinburgh's Charlotte Square and is owned by the National Trust for Scotland. It is used to host Cabinet meetings as well as functions for guests and dignitaries.
The complaints about Salmond were made in January, only weeks after Sturgeon approved the new procedures for dealing with harassment complaints.
Salmond is also seeking judicial review of the Scottish Government's of a new complaints procedure that was introduced by the SNP administration in December.
The announcement came after Sturgeon came under intense criticism over her claim that the SNP could not suspend Salmond, despite her government reporting the complaints to the police.
She said the complaints were being investigated by the Scottish Government, which was unable to share details of the complaint or the names of Salmond's two accusers to third-party organisations.
His decision to quit also comes amid growing tensions between Salmond and Sturgeon, his former protegee, who has refused to intervene in the case.
Salmond has pointed the finger at her Government for leaking details of the complaint. Senior Nationalist insiders fear a civil war between members supporting Salmond and Sturgeon.