Walker, who was convicted of a string of domestic abuse charges, bowed to pressure and stood down from his position as MSP for Dunfermline on Saturday.
Liberal Democrat leader Willie Rennie has urged the SNP to commission an independent inquiry into Walker’s selection as a candidate for the party, while Labour wants the party to publish all the information it possessed on any allegations against him.
Campaigners and the majority of MSPs had been calling on the 71-year-old to step down since he was found guilty of 23 domestic abuse charges against his three ex-wives and a step-daughter. He was found guilty following a trial at Edinburgh Sheriff Court last month. He carried out the attacks against between 1967 and 1995.
The former SNP MSP, who was ejected from the party when the allegations surfaced, denied the charges. He is due to be sentenced later this month.
Mr Rennie said: “Removing Bill Walker from the parliament was the priority.
“But now that he’s gone I am requesting the SNP commission an independent-led inquiry into these events and their internal processes.”
He added: “The SNP need to accept responsibility for Bill Walker taking a seat in our Parliament.
“No party is immune from unsuitable people seeking public office using their platform, but processes need to be robust to root them out when it is clear who they are.
“The SNP had been told of the allegations about Bill Walker but allowed him to continue to represent them on the [Fife] council, as a candidate and ultimately in our parliament.
“We need to know how this was allowed to happen, why a decision was made not to properly investigate the allegations and who made the decision.”
Conservative leader Ruth Davidson said: “Now Bill Walker has been convicted and has finally stood down from parliament, attention must turn to why he was put up for election in the first place.
“There are difficult questions here for the SNP to answer.”
An SNP spokesman said the party’s president, Ian Hudghton MEP, carried out a review of its assessment procedures following Walker’s expulsion last year.
“As a result, changes have since been made to our internal procedures to ensure that in future any relevant information about a potential candidate – whether backed by hard evidence or not – is passed to the election committee,” he said.
“In regard to Mr Walker, the investigation conducted by a member of staff at SNP HQ did not find any evidence of any complaint in law or legal proceedings into domestic violence by Mr Walker, and the inquiry was then closed.
“It is, of course, the case that knowing what is now known about Bill Walker, we all wish that he had been prevented from becoming an SNP candidate. Indeed, as soon as the evidence of the charges he was convicted of came to light, he was expelled from the party.”
As a constituency MSP, Walker’s resignation means a by-election will be held for the Dunfermline seat in the coming weeks.
A spokeswoman for the Scottish Parliament said Presiding Officer Tricia Marwick has received the resignation, effective from today.
She added: “There will be a by-election and Ms Marwick will consider when that will take place and notify the political parties in due course.”