The Oscar-winning actor was with wife Betsy Arakawa in Santa Fe when the incident occurred on Tuesday afternoon, reports the Albuquerque Journal.
Santa Fe police sergeant Andrea Dobyns said that the man complained of being slapped by Hackman in a phone call at around 1.30pm local time. Both Hackman and the complainant, who had no visible injuries, were then questioned by police.
Hackman said that the complainant had approached the couple in an aggressive manner and directed foul language at his wife.
"It looked like Mr Hackman did this purely out of self-defence to protect himself and his wife," Sergeant Dobyns told E! Online.
Charges are not expected to be filed against the French Connection and Royal Tenenbaums star.