He added that there was a possibility the resumption of the season, which was suspended on March 9, could be preceded by Italian Cup matches one week earlier.
"Italy is starting up again and it's only right that football can also get going," he told reporters after a conference call with representatives from the Italian Football Federation (FIGC).
Serie A will be the fourth of Europe's top five domestic leagues to restart following the Bundesliga, which resumed on May 16, and La Liga and the Premier League, which are also due to get under way again in June.
The French League ended the season last month with Paris St-Germain named Ligue 1 champions with 10 games left.
Spadafora said the Italian government approved health and safety measures suggested by the FIGC plus a backup plan in case the league had to be stopped again. "In the light of this we can say that the championship can start again on June 20," he said.
There are 12 rounds of matches still to play, plus four outstanding games from previous rounds.
Juventus, chasing a ninth successive Serie A title, lead that table with 63 points from 26 games, one point ahead of Lazio who are unbeaten in their last 21 league games.
Inter Milan are a further eight points behind in third with one game in hand.
The Italian Cup still has the second legs of the Napoli-Inter and Juve-AC Milan semi-finals to play plus the final.
Spadafora said one possibility was to play the semi-finals on June 13 and the final three days later.
One of the conditions of the Serie A league restarting was that the coronavirus testing it will entail "should not affect the general needs of all Italian citizens," he added.
Spadafora also said if there are signs of a new increase in coronavirus cases in general the league may have to stop again.
"The restart of football is a message of hope for the whole country. I am happy and satisfied," said FIGC president Gabriele Gravina, adding that he also hoped the lower tiers Serie B and C plus the women's Serie A could also restart. (Reporting by Angelo Amante in Rome and Elvira Pollina in Milan; Writing by Brian Homewood; Editing by Ken Ferris)