At the end of Quilmes' 1-1 draw at home to Atletico Rafaela on Friday, Vival went into the stands and landed three punches on the fan, a club member he said had been insulting him.
"I'm not a violent person but I realise that's not the image a trainer should give," Vivas said.
Club President Anibal Fernandez told Radio La Red that Vivas had gone too far.
"He came to my office and handed in his resignation. He knows he put his foot in it and realises the situation is uncomfortable for the team and regrets his actions," he said.
"We're trying all means to curb violence (in football in Argentina) and he can't commit such an error," added Fernandez, a senator for the ruling FPV party.
The 44-year-old Vivas, who took charge this season in his first job as a head coach, leaves Quilmes in 16th place in the Inicial championship with 14 points from 12 matches.
He began and ended his playing career at Quilmes and also turned out for Boca Juniors, Lugano, Arsenal, Celta Vigo, Inter Milan and River Plate, as well as the Argentina national side.
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