Meanwhile, anti-government demonstrations went on in the capital, and one protester was shot dead.
Demonstrators closed roads, including those leading to a major commodities port in southern Iraq.
A special judicial committee was formed to investigate demonstrator deaths.
On Friday, Abdul-Mahdi announced his plans to resign after violence broke out overnight across several cities and security forces killed at least 45 demonstrators over a 24-hour period.
The country's top Shiite cleric, Ayatollah Ali Sistani, suggested that lawmakers reconsider their support for Abdul-Mahdi's government.
Parliament approved the resignation Sunday without a vote, according to four lawmakers in attendance.
Lawmakers acted on the legal opinion of the federal Supreme Court because existing laws do not provide clear procedures.
"According to the federal court's interpretation, there is no need to vote," lawmaker Sarkwat Shamsedine said during the session.
Lawmaker Mohamed al-Daraji made the reference to a black hole in the law.
After the approval, Parliament Speaker Mohamed a-Halbousi asked President Barham Salih to nominate a new prime minister.
The constitution requires parliament's largest bloc to name a candidate for the premiership within 15 days.
Then the prime minister-designate has 30 days to form a government.
"Iraq has a historic chance to form a strong government, a government free from outside interference," said Ahmed al-Mayali, an Iraqi political analyst.
Gallery: Iraqis continue protests after official's resignation
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But officials and experts warned of a potential political crisis because the question of which coalition constitutes the largest bloc is unresolved.
Abdul-Mahdi's nomination as prime minister was the product of a provisional alliance between parliament's two main blocs -- Sairoon, led by cleric Moqtada al-Sadr, and Fatah, which includes leaders associated with the paramilitary Popular Mobilization Units headed by Hadi al-Amiri.
In the May 2018 election, neither coalition won a commanding plurality that would have enabled it to name the premier alone.
To avoid political crisis, Sairoon and Fatah forged a precarious union.
Salih began making rounds with different political blocs to reach a consensus, one lawmaker who requested anonymity in line with regulations said.
Two Iraqi officials also said that Iranian Gen.
Qassim Soleimani, the head of Iran's elite Quds Force and the architect of its regional security apparatus, arrived in Baghdad and met with key officials.
"It is expected that not just Soleimani but other usual brokers of the prime minister candidate will be active from now on," said one official who spoke on condition of anonymity because of regulations.
"But for sure no candidate will go through without the blessing of Najaf."
Najaf is the seat of Iraq's Shiite religious authority.
The possibility of Sairoon and Fatah re-committing to an alliance over the selection of the premiership was "the strongest scenario," Shamsedine said.
But the leaders of the two blocs have publicly split over the prime minister's resignation.
In a statement last month, al-Sadr vowed to never work toward a political consensus with al-Amiri again after the latter apparently backtracked from an agreement to subject Abdul-Mahdi to a no-confidence vote in parliament.
Officials also are grappling with the political preferences of outside powers such as the United States -- which was heavily involved in the political backroom deals ahead of Abdul-Mahdi's appointment last year -- and Iran.
Iran has played a significant role in Iraqi politics since Tehran threw its weight behind the militias.
But that support may have backfired.
Demonstrators in majority-Shiite cities such as Najaf and Karbala have stormed or burned Iranian consulates and defaced posters of Iran's supreme leader, Ayatollah Ali Khamenei.
Iran is governed by a Shiite theocracy and has used military and financial largesse to help shape Iraq's political landscape.
"We really hope that Iran this time will leave the Iraqi people to choose their own fate," said Naji al-Saadi, a member of the pro-al-Sadr Sairoon bloc.
According to Mayali, "Iranian leverage is not on the level it was before the protests."
Iranian officials, the political analyst said, "are no longer free to form a government that is close to Iran."
But even as lawmakers paid lip service to the need for change, they conceded that a new leader would inherit a restive and fractured nation that is reeling from a security crackdown that appears to have only hardened protesters' resolve.
In Baghdad's historic Rasheed Street on Sunday, security forces fired live ammunition to prevent crowds from breaching concrete barriers near Ahrar Bridge that leads to parliament and other government buildings.
One protester was killed and 10 wounded, according to security and medical officials who spoke on condition of anonymity.
Hundreds of anti-government demonstrators, including students and teachers, also took to the streets in the southern oil-rich city of Basra.
They wore black clothes to mourn protesters killed in Najaf and Dhi Qar provinces in recent days.
The new investigative committee was formed to hear cases from the city of Nasiriyah, which has seen the most fatalities from live ammunition used by security forces.
The committee issued an arrest warrant against Lt.
Jamil al-Shammari on charges of issuing orders that led to the killing of demonstrators, according to Iraqi state TV.
Abdul-Mahdi recently withdrew al-Shammari from overseeing security matters in the southern city after the bloodshed.
Unlike elsewhere in Iraq, Basra demonstrators have routinely targeted the country's economic interests.
Demonstrators staged a sit-in and cut roads to the West Qurna 1 oil field, operated by Exxon Mobil.
The field, among the country's largest, produces over 450,000 barrels of oil per day.
A senior oil ministry official said the protests have not yet affected crude production.
Protesters continued to block roads to the country's main Persian Gulf commodities port in Umm Qasr.
Port officials previously said trade activity had been cut by 50% as a result.
Also Sunday, unknown attackers in Najaf torched the Iranian Consulate, which was empty.
It was the second time the building had been set ablaze in recent days, after an earlier fire started by protesters who stormed the structure.
The grassroots protest movement, which began Oct.
1, has posed the most serious challenge to Iraq's political order since the U.S.-led invasion in 2003.
More than 430 demonstrators have been killed, a human-rights official said Sunday.
He spoke on the condition of anonymity because he was not authorized to release casualty figures to the media.
In Baghdad, protesters gathered in Tahrir Square, the epicenter of the movement, to reiterate calls for a complete overhaul of the sectarian political system.
Hundreds of university students skipped classes to attend.
"First, we want a country.
Second, we want all of them out.
No one stays.
They are all thieves," said a demonstrator who gave her name as Umm Zaynab, as protesters chanted anti-government slogans.
Information for this article was contributed by Samya Kullab and Murtada Faraj of The Associated Press and by Erin Cunningham and Mustafa Salim of The Washington Post.
P/HADI MIZBAN Anti-government protesters in Baghdad clean Rasheed Street which was damaged Sunday in clashes between the demonstrators and security forces.
A Section on 12/02/2019