Turnbull pre-emptively held a leadership ballot on 21 August, contested by Turnbull and Home Affairs Minister Peter Dutton, which Turnbull won by 45 votes to 38.
Even as Turnbull rejected any retribution against ministers suspected of supporting Dutton, Assistant Minister to the Prime Minister James McGrath, Minister for International Development and the Pacific Concetta Fierravanti-Wells, and Assistant Treasurer Michael Sukkar confirmed within hours that they had offered their resignations.
His decision will trigger a by-election in Turnbull's seat of Wentworth, potentially as soon as October 6.
Senior Liberals said Mr Sharma was a frontrunner until Mr Bragg indicated he was interested.
"Australians expect that of their parliament and I'm pleased that's the case", Morrison told reporters when asked about comments from former Prime Minister Tony Abbott, a leader of the conservative faction that ousted Turnbull, that the "era of the political assassin is over".
Morrison was sworn in as prime minister Friday after winning a Liberal Party leadership challenge against incumbent moderate Malcolm Turnbull.
Morrison has a one-seat majority, but will face a by-election when Turnbull resigns from parliament, which ABC news reported could be as early as later this week.
Morrison has been dubbed the "accidental prime minister" because he had no plans to nominate until Thursday, when Turnbull declared he would not recontest his job.
"To say "oh well I'm not the prime minister anymore ho hum, you guys might lose government" - the government that he led - I think people are really disappointed with him about that".
It was part of Mr Morrison's olive branch attempt to appease those who took down Malcolm Turnbull last week, and begin mending the party's wounds.
But he said he didn't want to dwell on the "shocking and shameful events of last week - a pointless week of madness that disgraced our parliament and appalled our nation".
In a speech in Sydney on Monday Mr Abbott said Mr Morrison had strong character and expected him to install a "sharper difference" between the Liberal and Labor parties ahead of the next federal election.
"I thank Jennifer Westacott, the staff and members for the opportunity".
Mr Turnbull comfortably held the blue-ribbon seat of Wentworth with a margin of 17.7 per cent at the 2016 election.
Christine Forster - the sister of arch-conservative Tony Abbott, who helped orchestrate Turnbull's demise - has already put her hand up to replace Turnbull in the plum seat.
Ms Forster has been a Liberal councillor for the City of Sydney for the past six years and was a leading figure in last year's successful same-sex marriage campaign.