On Thursday, Abadi announced a new series of public works initiatives for the southern city of Basra, hit by a wave of deadly protests last week.
The move comes just two days before a key parliamentary meeting that may allow the premier's political rivals to form a government without him.
Anger in Basra flared early last week over a growing health crisis, after more than 30,000 people were hospitalized by pollution in the city's water supply.
Protests continue to grow in Basra, where violent demonstrations against poor services and government corruption have been raging for months.
The outgoing Iraqi prime minister also called on political blocs to speed up their nomination of the three leadership posts of parliament speaker, president and prime minister.
"I did not and will not request the post of prime minister in the second term", Anadolu Agency quoted Abadi as he addressed a news conference in Baghdad.
This could also be a big blow to the United States, which has long tried to retain effective control over Iraq's government, and has driven the previous governments' formations virtually every election.
Mr Sadr's election list came in first in the polls, winning 54 seats in the 329-seat legislative body, while Mr Abadi's list came in third, with 42 seats.
Iraqi Prime Minister Haidar al-Abadi has indicated he may not seek a second term amid power struggles in the country, although observers say it was unclear if he was actually giving up any claim to power.