The Champions League finalists were winning 2-0 in injury time following goals from Francesco Pio Esposito and Alessandro Bastoni, while the Argentine outfit were down to NINE men after seeing Lucas Martinez Quarta and Gonzalo Montiel sent off in the second half.
Montiel had been sent off minutes after Bastoni's injury-time stunner.
And Acuna looked to take his frustrations out on Dumfries after the Dutchman left a heavy challenge on him just seconds before.
Dumfries did not let the taunts get to him and even gave some back, but when the full-time whistle came from referee Ilgiz Tantashev it kicked off again.
As the 29-year-old jogged off the pitch at Lumen Field in Seattle, he was pursued by 33-year-old Acuna - all while coming under a shower of missiles from the River supporters.
Players from both sides rushed over to the corner of the pitch to get in the middle of the pair, with both River subs and Inter head coach Cristian Chivu included in a group who held Acuna back before wrestling him to the ground.
The result confirmed Inter's progression into the last 16 as group leaders, which has set them up for a mouth-watering tie against fellow South American club, Fluminese.
Meanwhile, River were sent crashing out of the tournament after finishing a point behind Mexican outfit Monterrey, who fought their way to two draws against Inter and River before thrashing Japanese giants Urawa Reds 4-0.