Prime Minister Malcolm Turnbull earlier called for Wilson to resign, saying he was surprised he had not already done so.
"I am surprised that he has not resigned. Clearly, given the outcome of the case, the lawsuit and prosecution, he should resign," the prime minister told reporters on Kangaroo Island in South Australia on Wednesday.
The 67-year-old was in May found guilty of failing to report to police the historic sexual abuse of two altar boys by a pedophile priest, after a landmark magistrate-only trial in Newcastle Local Court.
Wilson said he plans to file an appeal and will not be resigning while the court process is ongoing.
"I am conscious of calls for me to resign and have taken them very seriously. However, at this time, I am entitled to exercise my legal rights and to follow the due process of law," he said in a statement.
"Since that process is not yet complete, I do not intend to resign at this time. However, if I am unsuccessful in my appeal, I will immediately offer my resignation to the Holy See."
Magistrate Robert Stone on Tuesday sentenced him to 12 months' imprisonment with six months non-parole over the offence, saying Wilson's main motivation had been to protect the church.
"In this case the offender is a senior figure in one of the most respected institutions in our society," Mr Stone said.
However, in a decision that prompted sighs in the packed public gallery, the magistrate ordered that Wilson be assessed for an order allowing him to serve his sentence in the community.
Wilson, who said nothing to media as he left court, will return to court on August 14 when the assessment has been completed.
In finding him guilty, the magistrate accepted that Peter Creigh and another altar boy in 1976 told him they'd been abused by priest James Fletcher but Wilson did nothing.
His offence relates to the period between 2004 and 2006, when Fletcher was charged with child sex offences.
The magistrate found Wilson had at that stage obtained the level of belief needed to report what he'd heard to authorities.
Fletcher was found guilty of child sexual abuse in 2004 and died in jail of a stroke in 2006.
