Earlier on Friday, Attorney-General Gali Baharav-Miara penned a letter to Levin, calling on him to convene the committee.
The new bill is based on a joint proposal by Justice Minister Yariv Levin and Foreign Minister Gideon Sa'ar and is expected to pass into law by the end of February.
The AG wrote that it was not clear why Levin waited until the days before the deadline to air the allegations, some of which arose already in 2023.
The judicial selection process is complex, involving input by judges, the Israel Bar Association, and Knesset.
Foreign Minister Sa'ar and Justice Minister Levin propose new judicial compromise, offering concessions on Supreme Court appointments and Basic Laws.
The law in question would grant the coalition control of the Judicial Selection Committee.
The movement further stated it had requested the court to sanction Levin until he abides by the ruling and asked that a discussion be conducted on the subject.
The Justice Minister must name Chief Justice candidates by the end of the day.
The bill proposed to change the process to elect an ombudsman for Israel's court system.
This follows the petition submitted by the Movement for Quality government, which claims that the Justice Minister refuses to convene the Judicial Selection Committee.