Since May 2024, the position of Ombudsman of the Judiciary has remained vacant.
The new proposal also includes a mechanism to prevent a stalemate in high court appointments.
"This is a sad day for democracy and a sad day for our judicial system," the justice minister said in a statement.
The Judicial Selection Committee convenes to appoint Israel's new chief justice.
The bill proposal, by Likud MK Osher Shkalim, is one of a number of proposals targeting Kan, which are being promoted by Communications Minister Shlomo Karhi.
Earlier on Friday, Attorney-General Gali Baharav-Miara penned a letter to Levin, calling on him to convene the committee.
The ultra-Orthodox IDF draft has come to a head as the Israel-Hamas war has seen reservists completing hundreds of days of duty. IDF says it needs more manpower to deal with the new security reality.
The final tally was 53-45 against the proposal. However, the coalition has attempted to come up with an alternative committee.
Israel has been without a chief justice since October 2023, when former chief justice Esther Hayut’s tenure ended.