Former Labour MP Ian Austin said Sir Keir would need to launch a thorough internal investigation if he was serious about "tearing out the poison" of anti-Semitism that had blighted the party under his predecessor.
It came as a leaked Labour report concluded that efforts to tackle the issue were hindered by hostility towards Mr Corbyn.
It said an "abnormal intensity of factional opposition" to the former leader "inhibited the proper functioning" of the party and its complaints procedure.
The document was intended to be submitted to the ongoing Equalities and Human Rights Commission (EHRC) inquiry into anti-Semitism in the Labour Party, but it is now understood it will not be sent.
Mr Austin was the MP for Dudley North before quitting the party over the rise of anti-Semitism under Mr Corbyn's leadership.
He said the report was unreliable as it had attempted to "shield" the former Labour leader and his supporters from any blame.
"The real issue is how what had previously been a fringe issue in dark corners developed so much under Jeremy Corbyn," he said.
"Why did racist cranks and conspiracy theorists think Labour under him was their party and what is his responsibility for normalising this poison?
"Tearing out the poison won't just mean sorting outstanding cases, but subjecting those swept under the carpet under Corbyn to the new process Keir Starmer has promised as well so all the racists can be booted out."
The report’s conclusions clash with complaints of whistleblowers, who told BBC Panorama last July there had been political interference in the complaints process from the top of the party under Mr Corbyn.
At the time their claims were strongly denied by Labour, which accused the BBC of bias.
Sir Keir has launched an internal inquiry into the leak, having previously vowed to address the "disgrace of anti-Semitism in our party as soon as possible".
Mr Corbyn's time as Labour leader was dogged by claims that officials did not act strongly enough against claims of anti-Jewish racism.
Gideon Falter, chief executive of Campaign Against Antisemitism, said the leaked report showed Labour had "invested in a desperate last-ditch attempt to deflect and discredit allegations of anti-Semitism".