Turnbull, who has previously called on Wilson to resign, escalated his criticism of Wilson remaining in office.
Wilson's defence barrister Ian Temby QC told a sentencing hearing last month that imprisonment is likely to worsen his client's many chronic illnesses and may put him at risk of assault from fellow inmates.
While child survivors of sexual abuse are welcoming Malcolm Turnbull's intervention, some within the Catholic Church are questioning it.
Mr Turnbull said "the time has come for the Pope to sack" Wilson because it was "clear that he should resign".
Wilson, who was sentenced to a year's detention this month, has stepped aside as Adelaide archbishop, but has not resigned, insisting he will do so only if he is unsuccessful in an appeal.
"If he doesn't have the decency to resign then his superiors in the church should take action", Fairfax media reported Mr Shorten as saying.
"While the archbishop is exercising his constitutional right to appeal his conviction, his tenure as archbishop of Adelaide has been compromised".
The Adelaide Archbishop has said that he will appeal his conviction and that he would resign if that appeal failed.
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"Since that process is not yet complete, I do not intend to resign at this time".
Pope Francis (C) during the funeral of Cardinal Jean-Louis Tauran, St. Peter's Basilica, Vatican City, 12 July 2018.
Instead, Wilson's legal team urged the a good behaviour bond for the offence that carries a maximum two-year jail term.
Magistrate Robert Stone in Newcastle Local Court sentenced Wilson to imprisonment with a non-parole period of six months.
Turnbull is a former lawyer who was born a Presbyterian but converted to Catholicism - his wife's religion - in 2002.
"He should have resigned and the time has come for the Pope to sack him", Turnbull said at a press briefing. Each bishop is appointed by the Pope personally so that the authority to compel a bishop to resign does not belong to any other bishop or even the collective of Australian bishops.
Wilson was the bishop of Wollongong before being elevated to Archbishop of Adelaide.
The Australian government is expected to issue an apology in October to the victims on behalf of the state.
