The opposition Democratic Party had threatened to impeach Han if he does not immediately appoint three justices to fill the vacancies at the Constitutional Court.
Parliament voted in favour of three nominees on Thursday, but they have yet to be formally appointed by Han.
The court is trying the impeachment of President Yoon Suk-yeol over his December 3 declaration of martial law.
"It has become clear that Prime Minister and Acting President Han Duck-soo does not have the qualification or the will to safeguard the constitution," Democratic Party floor leader Park Chan-dae said in a statement.
Acting President Han Duck-soo will be replaced by the finance minister if he's impeached. (AP PHOTO)
The motion was introduced to parliament on Thursday and must be voted on within 24 to 72 hours.
It cited a range of actions by Han as grounds for his impeachment, including his veto of a special prosecutor bill to investigate alleged wrongdoings by the first lady.
If Han is impeached, the finance minister will assume the acting presidency.
The Democratic Party has majority control of parliament, but there is disagreement between the parties and some constitutional scholars over whether a simple majority or a two-thirds vote is needed to impeach the acting president.
Han said earlier on Thursday he would not appoint the justices until political parties reached agreement on the appointments, because for him to do so without political consensus would harm the constitutional order.
Two of the proposed appointees for the Constitutional Court approved on Thursday were nominated by the Democratic Party and one by Han's ruling People Power Party.
The ruling party objected to that breakdown of candidates, saying it had not agreed to it.
Han has been under pressure to make the appointments, but political parties have disagreed on whether he has the authority to do so as acting president.
President Yoon Suk-yeol was impeached by South Korea's parliament on December 14. (AP PHOTO)
The court will hold its first hearing on Friday in the trial to decide whether to remove Yoon or reinstate him.
Under the constitution, six justices must agree to remove an impeached president, meaning the current justices must vote unanimously to remove Yoon.
The court has said it can deliberate without the full nine-member bench.
Yoon, who was impeached by parliament on December 14 in a vote joined by some members of his centre-right party, had not submitted legal papers as requested by the court as of Thursday, court spokesperson Lee Jean told a media briefing.
On Wednesday, he did not respond to the latest summons for questioning in a separate criminal investigation.
Investigators have sent another summons on Thursday for him to appear on December 29.
Yoon's repeated defiance has sparked criticism and calls from the opposition for his arrest.
