Abbott threw his support behind Morrison and Liberal Deputy Leader Josh Frydenberg, saying their leadership marked an "important reset" for the Government.
But Morrison could not restore the former prime minister to Cabinet, despite hints he might have been promoted under a Dutton government.
Morrison, 50, is seeking to unite the Government ahead of elections due by next May. He named Marise Payne to replace Julie Bishop as Foreign Minister and promoted Christopher Pyne to oversee defence. Mathias Cormann will stay in his role as Minister for Finance and Dutton retained a reduced home affairs portfolio.
The Coalition now trails the Labor party by a 12-point margin, 44 per cent to 56 per cent, compared with a two-point gap this month, according to the Newspoll published in the Australian.
Labor leader Bill Shorten is now seen as the preferred prime minister for the first time since February 2015.
There are growing calls for the Liberal Party to change the rules to make it more difficult to bring on a spill.
Nick Greiner, the federal Liberal president, suggested the party would benefit from doing so. "I'm not opposed to that idea, I think it's an obvious directional thing," he told Sky News.
"I don't think it's on the top of Morrison's list, nor should it be, but I think the organisation, not surprisingly, would be happy with some sort of model.
"There are already members of the parliamentary party advocating it and Mr Morrison does not have a tin ear, but yes certainly I think it's something that ought to be on the agenda."
But former Liberal leader John Howard has rejected such calls.
"I don't think changing the rules is a good idea," he said, according to Fairfax Media. "What's the point of bringing in rules if, in any event, they can be set aside?"
Ironically enough, Howard was the last Australian prime minister to serve a full term. That was over 10 years ago.
- additional reporting Bloomberg