Mr Macri said the two countries should not let their bitter football rivalry spill over into the Olympic Games.
Brazilian fans have been loudly supporting any athlete competing against their southern neighbours.
Argentine fans have often responded with disrespectful chants against the Brazilians.
As the former president of Argentina's popular football club, Boca Juniors, Mauricio Macri is familiar with the problems caused by bitter rivalries in sports.
In a message posted on Facebook, he said he enjoyed supporting his country and his club.
But he warned that things were getting out of control at the Rio Olympics and could end badly.
"The sports rivalry between Argentines and Brazilians has turned into a verbal confrontation that could end badly," said Mr Macri in the message.
"Neither we, Argentines, nor the Brazilians are like that."
Brazilian fans have been booing Argentine athletes in any competition they take part in.
Tennis player Juan Martin del Potro was constantly booed by the Brazilian crowd during a match against his Portuguese rival, Joao Sousa.
Two fans ended up being expelled from the stands for exchanging punches.
Mr Macri quoted the Argentine basketball team captain, Luis Scola, who said during the week: "I don't want Brazil to lose. I just want them to lose when they play Argentina. It seems silly to me to sing chants against a team that is not even on the court."
The Argentine president's message came ahead of a much-awaited men's basketball match against the two South American rivals on Saturday afternoon in Rio.
Despite the emotional and thrilling match, which Argentina narrowly won 111 to 107, supporters from both sides have been praised for dancing and cheering their teams throughout.