The move, announced in a government decree, will also affect the Six Nations rugby international between Italy and England, due to be played in Rome on March 14, and the Champions League soccer match between Juventus and Olympique Lyonnais on March 17.
All schools and universities were closed on Wednesday to try to slow the spread of the coronavirus in Italy where there have been more than 3,000 cases and the death toll has reached 107.
The ban on the public watching sports events, which had already been in force in three regions in the north but has now been extended nationwide, will last until April 3 at the earliest.
The decree said that "sports events and competitions of any order and discipline, carried out in any place, both public and private, are suspended" but could go ahead if there were no spectators.
It added that "sports associations and clubs, through their own medical staff, are required to carry out appropriate checks to limit the risk of spreading (the coronavirus) among athletes, technicians, managers and all accompanying persons participating."
Until now, Serie A has preferred to call off games in the affected regions rather than play them in empty stadiums and 10 matches have been postponed as well as this week's two Coppa Italia semi-final ties.
However, after a meeting of club and Serie A representatives earlier on Wednesday, Inter Milan chief executive Giuseppe Marotta said that playing without fans could be the only way of finishing the Serie A season.
"Playing behind closed doors could be the only way to complete the championship in the light of the emergency and the restrictions that the government is rightly adopting," he said as he left the meeting.
"The aim is to finish the championship as normally as possible, without creating a competitive imbalance, but as you can see the scenario is constantly changing."
Marotta suggested that the six matches which had been postponed last week -- including the heavyweight Juventus v Inter clash -- could be rescheduled for the coming weekend.
Six Nations organisers have postponed the match between Ireland and Italy in Dublin on March 7 but said on Monday that all other games would go ahead, including Italy's game against England in Rome, which was not affected by the original ban in the northern regions.
The organisers and the Italian rugby federation could not immediately be reached for comment.
UEFA has said it is in constant contact with local authorities and decisions over whether games should be played behind closed doors, or postponed, would be taken at the last minute.
