“I resigned and, as a purely political act, I asked the board to do so too, but no one did,” Mr Tavecchio said in a news conference.
The former president complained of the “huge speculation” which ultimately forced him to step down over the World Cup failure.
Mr Tavecchio fired Italian coach Gian Piero Ventura last week after the four-time world champions lost to Sweden in a playoff to reach next year’s tournament in Russia.
He was under intense pressure, with Italian Olympic Committee (CONI) President Giovanni Malagò saying he should step down, and many commentators arguing that Mr Ventura’s incapacity as coach was not the only reason behind the disappointing loss.
Mr Malagò said on Monday that the FIGC should now be placed into the hands of independent commissioners.
Mr Tavecchio, however, rejected this notion.
The shunned president will remain in charge of the day-to-day running of the FIGC until the new election, which should be called within 90 days.