He resigned at the weekend from his Dunfermline seat following increasing public outrage over his conviction for a string of domestic abuse charges,
The Liberal Democrats have called on the SNP to set up an independent inquiry into Walker's selection as a candidate, while Labour wants the party to publish all the information it possessed on any allegations against him.
Campaigners and most MSPs had been calling on the 71-year-old to step down since he was found guilty of 23 domestic abuse charges last month. He carried out the attacks, against his three successive wives and a stepdaughter, between 1967 and 1995.
Walker, who denied the charges, is due to be sentenced later this month. He was ejected from the SNP when the allegations surfaced.
Scottish LibDem leader Willie Rennie said: "Removing Bill Walker from the parliament was the priority. But now he's gone I am requesting the SNP commission an independent-led inquiry into these events and their internal processes.
"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 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."
Labour's Graeme Pearson said: "We need to know why the warnings about Bill Walker's violence were not heeded by some of the most senior members of the SNP.
"The SNP needs to be open and upfront about what happened and why it didn't block Bill Walker as a candidate. We know a member of Bill Walker's family told (SNP deputy leader) Nicola Sturgeon's staff about the sort of man he is.
"The SNP had the knowledge to stop a now convicted wife-beater from entering our Parliament. It didn't. We need to know why."
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 been made to our internal procedures to ensure 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 closed. Knowing what is now known about Bill Walker, we all wish he had been prevented from becoming an SNP candidate."
Walker's resignation means a by-election will be held for the Dunfermline seat in the coming weeks.
Yesterday, his ex-wife Anne Gruber said his decision to stand down "comes as a great relief to me and, I'm sure, all of his victims".