"I will not allow Israel to fall into anarchy," he added.
Opposition parties say Mr Edelstein was himself subverting democracy by refusing to convene parliament to discuss the issue.
After elections earlier this month Likud and its right-wing allies have 58 seats in the 120-seat Knesset, three short of a majority, while 62 MKs are opposed to Mr Netanyahu continuing as prime minister.
Mr Edelstein had argued a new speaker would be an obstacle to attempts to form a national unity government between Likud and Blue & White, the main opposition alliance.
But in a statement on Tuesday, Blue & White leader Benny Gantz distanced himself from the prospect of joining a Netanyahu-led government.
The speaker's resignation only takes effect in 48 hours, meaning the Knesset is unlikely to be able to pass laws until next week.