One of the highest-ranking church officials to be convicted on the charge, the Adelaide archbishop was found guilty in May of concealing abuse by a notorious paedophile priest during the 1970s and was sentenced this month to 12 months' home detention.
Wilson, 67, has stepped down from his duties but resisted calls to resign pending an appeal of his conviction, prompting Turnbull to call on the Vatican to intervene.
"He should have resigned, and the time has come for the Pope to sack him," Turnbull told reporters in Sydney.
"There are many leaders that have called on him to resign, it is clear that he should resign, and I think the time has come now for the ultimate authority in the church to take action and sack him."
Turnbull was speaking ahead of a meeting Thursday with senior Catholic officials.
Wilson was found guilty by failing to report allegations against paedophile priest Jim Fletcher.
The archbishop has long denied the charges and his legal team made four attempts to have the case thrown out, arguing the he suffered from Alzheimer's and so should avoid trial -- even though the diagnosis did not prevent him retaining his position in the church.
Newcastle Local Court magistrate Robert Stone found Wilson guilty of concealing a serious indictable offence of another person, concluding his primary motive was to protect the church.
In sentencing, Stone said "there is no remorse or contrition showed by the offender".
Like elsewhere in the world, Australia has been plagued by accusations that the Catholic Church ignored and covered up child abuse.
A national inquiry into the issue was ordered in 2012 after a decade of pressure to investigate widespread allegations of institutional paedophilia.
The royal commission -- which ran for five years -- spoke to thousands of victims and heard claims of abuse involving churches, orphanages, sporting clubs, youth groups and schools.
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