Ian Blackford has been urged to explain why no action was taken after sexual harassment allegations were made against Patrick Grady four years ago.
Scottish Conservative MP John Lamont has written to the SNP's Westminster leader asking why his party hasn't suspended the Glasgow North MP while an investigation is underway.
The Daily Record exclusively revealed that Grady was accused of inappropriately touching a young staffer in a London pub 2016.
Following the revelations a formal complaint was submitted last week.
Grady has now resigned as the SNP 's chief whip at Westminster while an investigation is conducted.
At the weekend it was revealed, in a Sunday newspaper, that former First Minster Alex Salmond is believed to have warned Sturgeon about sexual harassment allegations against Grady three years ago.
Lamont also quizzed the Ross, Skye and Lochaber MP on what steps his party are taking to combat sexual harassment in the SNP group.
In his letter Lamont, MP for Berwickshire, Roxburgh and Selkirk, asked Blackford three questions including whether the SNP will now suspend Grady and an unnamed female MP also accused of sexual misconduct.
It states: "1. In light of the claims outlined above, will you commit to immediately suspending both Patrick Grady and the other as-yet-unnamed SNP MP from the SNP, subject to investigation. This will include from any spokesperson role that the as-yet unnamed MP may have?
"2. Before formal complaints were made, to what extent were you and your predecessor as SNP Leader aware of concerns about inappropriate behaviour by both Patrick Grady, and the other as-yet-unnamed SNP MP?
"3. What steps are you taking, as the Westminster leader of the SNP, to combat sexual harassment in the SNP MP group?"
He said: "The SNP's Westminster leader Ian Blackford and other senior figures have some serious explaining to do.
"We now know that party employees raised concerns about the chief whip's alleged sexual misconduct as long ago as 2017 — yet nothing appears to have been done.
"The suggestion that Grady was ‘protected’ by the party is chilling and must be urgently and honestly investigated; not the usual SNP cover up.
"This appears to go to the very top. If Blackford knew and did not act, what about Nicola Sturgeon and Peter Murrell?
"In my letter, I have asked a number of key questions about Grady and the other SNP MP who faces similar sexual misconduct allegations.
"One question is about whether these two MPs will be suspended by the SNP while these matters are investigated.”
An SNP spokesperson said: "The SNP has followed due process and taken appropriate action at every stage.
"The SNP National Secretary has now had a formal complaint notified to him and will investigate accordingly.
"But we will take no lectures from the Conservative Party who have a frankly appalling record when it comes to handling complaints - such as the decision to readmit two suspended MPs ahead of a vote of no confidence in Theresa May, refusing to suspend an MP arrested on suspicion of rape, and hushing up an investigation into a senior MSP until he was cleared in a process heavily criticised by Rape Crisis Scotland."
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