Mr Morrison has faced mounting pressure to step down from federal parliament following the findings of the robodebt royal commission.
The inquiry found coalition ministers, including Mr Morrison, ignored or dismissed concern about the legality of the scheme.
Mr Morrison has denied any wrongdoing.
The royal commission has recommended individuals face civil or criminal charges relating to the unlawful scheme but the names have not been published, instead featuring in a sealed section of the report.
Mr Dutton said former prime ministers needed to be treated with respect and Mr Morrison was the only one to decide whether he should step down.
"It's Scott's decision as to whether he stays or goes. He's a good representative of his local community and he'll make the decision that's right for him," the opposition leader told Sydney radio station 2GB on Thursday.
"Prime ministers past had a special place in the parliament and there's an expectation that people either serve out the full term or go to a by-election, depending on the circumstances."
Mr Dutton said those named in the royal commission report as being recommended for charges should be afforded the presumption of innocence.
"If you've done the wrong thing, and there's a finding of guilt, then you should be held to account for that and answer it and and cop the penalty," he said.
"But we don't find guilt before innocence, and this is not a judge and jury process but there are lessons to be learned and that's the situation that we're in at the moment."
Mr Dutton said the coalition had taken responsibility for wrong decisions it had made while in government that led to the robodebt scheme, which was declared unlawful.
Deputy Prime Minister Richard Marles said voters were right to be concerned about the actions of former ministers.
"The real villains here are not the public servants - it is the former government," he told ABC radio.
"The guilty party here is the Liberal Party.
"They are the ones that put in place the culture ... and made the decisions which enabled the appalling, gross maladministration that we saw in the handling of robodebt to impact the lives of half a million Australians."
Meanwhile, embattled consultancy firm PwC said it had returned almost $900,000 to the government for work it done out for a report into the unlawful scheme.
PwC acting chief executive Kristin Stubbins said it would have been inappropriate to retain fees it had received from the Commonwealth for work it had carried out.
"Like all Australians, we have found the contents of this report deeply distressing," she said.
"For PwC's part, we acknowledge the findings of the commission in relation to PwC's work for the Department of Human Services in relation to robodebt."
Gordon Legal, the firm that launched the class action lawsuit against the robodebt scheme, was considering fresh compensation claims following the commission's findings.