Bishop was eliminated on the first vote, setting up a showdown between Morrison and Dutton, which Morrison won 45 votes to 40, making him the new Liberal leader.
Mr Dutton, a conservative, then unsuccessfully challenged Mr Turnbull on Tuesday, but his narrow defeat only stoked further discord.
After a morning of chaos and multiple ministerial resignations, including that of Senate leader Mathias Cormann, Turnbull told his news conference that if he is ousted he will quit parliament - increasing the prospect of an early election.
He ruled out calling a general election in the near term but will still face an early electoral test, as Turnbull is set to resign from parliament, forcing a by-election in a Sydney seat that has been a safe seat for the Liberals.
Another Turnbull supporter, Darren Chester, apologised to voters and said they "deserve better than numerous things our Federal Parliament has served up in the past 10 years".
The leadership now looms as a battle between Mr Dutton and Treasurer Scott Morrison, who could run if Mr Turnbull steps aside.
He built his reputation in the party as a tough but effective operator when he took on the immigration portfolio from 2013-14, implementing the controversial "Operation Sovereign Borders" to stop asylum-seekers arriving in Australia by boat.
A harsher assessment of Morrison comes from the refugee lobby group "Love Makes a Way": "Given the certainty that Morrison will be portrayed as a devout Christian and a "moderate", it needs to be said that Scott Morrison has betrayed nearly every principle that is central to Christian political ethics".
"Australians will be rightly appalled by what they are witnessing in their parliament", Turnbull told reporters in Canberra.
He has a reputation as a power broker within the Liberal Party whose support has been crucial to previous party leaders.
Coupled with the government long trailing the opposition Labor party in opinion polls, the move sparked a revolt that quickly spiralled into calls for his head.
Ricciardo: "Bad blood" didn't force his move away from Red Bull
Verstappen a year ago signed up to Red Bull through 2020 and the Dutchman believes his current team-mate has not chosen the best place for his future.
"It was described as madness by many, and I think it's hard to describe it in any other way".
"Both underlined the strength and depth of our alliance and the unbreakable friendship between Australia and the United States", the spokesperson said.
He said the result meant West Australians could "sleep easy" because WA's GST deal would be "secure" under Mr Morrison's leadership.
"Australia. We owe you an apology".
Australia's next prime minister, Scott Morrison, is seen by some as an incongruous mix of a committed Christian who made his name through ratcheting up a refugee policy that many church groups have condemned as inhumane.
I don't think any Australian Prime Minister has used the word "love" more frequently in his public remarks.
They were either sent back to where they had come from or transferred to remote Pacific island camps.
China hopes the change of leader will thaw bilateral relations that have been chilled for months by Turnbull's crackdown on covert foreign interference in domestic politics and on industrial espionage.
After Mr Abbott was removed as prime minister, Mr Turnbull rewarded Mr Morrison's support with a promotion to the prized position of treasurer, where he handed down his first federal budget in 2016.
Shorten said the "cannibalistic behaviour" over the Liberal leadership was eating the government alive.