Patrick Grady, who was found by an independent panel to have touched and stroked the neck, hair and back of a colleague 17 years younger than him at a social event in 2016, received a suspension from Westminster for two days.
It was revealed over the weekend that he is now subject to a police investigation into his behaviour following a complaint from a third party.
The Metropolitan Police said it would make inquiries into the incident and contact the alleged victim before assessing what, if any, further action is required.
A spokesperson for the SNP confirmed on Sunday morning that Mr Grady, the former chief whip in Westminster, will be “stepping away from his party membership while the police inquiry continues.”
This will see him no longer sit as an SNP MP until the probe is completed.
The Glasgow North MP will instead sit as an independent.
SNP Westminster leader Ian Blackford had urged Mr Grady to consider his position after criticism of the party’s handling of the complaint.
The scandal has rocked the SNP after a leaked recording of an internal meeting of MPs showed Mr Blackford encouraging colleagues to give “as much support as possible” to Mr Grady.
Amy Callaghan, the East Dunbartonshire MP, was heard stating SNP MPs should be “rallying together” to support the MP.
Both later apologised but have faced accusations from the victim that they were not supported effectively.
The staffer called Mr Blackford’s statement a “non apology and a bit of a cop out”.
In an interview last week, Mr Blackford was asked six times if it was permissible for a “sex pest” to be an MP. He refused to directly address the question.
Nicola Sturgeon also faced pressure from opposition parties at First Minister’s Questions.
She issued a stinging rebuke of the behaviour of the Westminster group, saying the recording of the meeting was “utterly unacceptable” and showed MPs were more concerned for the perpetrator than for the victim.
The First Minister publicly apologised to the victim and offered to meet with them to say sorry in person.
In responding to questions, she added that the issues facing the SNP were not unique and “all parties” had faced similar issues.
She said: “All of us have lessons to learn.”
Scottish Conservative party chairman, Craig Hoy, said the MP should have been sacked long before this probe and accused the SNP of having “no backbone”.
He added: “The abuser is gone but the poison remains in the SNP and the party leadership should be ashamed of the way they have handled this scandal.
“Bullying Blackford must now face the repercussions he deserves and be sacked so victims have confidence that they’ll be taken seriously in the future.”
A Metropolitan Police spokesperson said: “On Wednesday June 22, police received an allegation of sexual assault that is said to have taken place in October 2016 at a commercial premises in Folgate Street, E1.
“The report was submitted online by a third party.
“Officers will now be making inquiries, including contacting the alleged victim, in order to assess what further action is required.”