The 32-year-old Barcelona forward had accused CONMEBOL of "corruption" after he was sent off against Chile during the third-place play-off during the tournament in Brazil, which ended last month.
Messi was angered by two incidents during the June-July Copa hosted by bitter rivals Brazil.
Argentina were denied two penalty claims in their 2-0 semi-final defeat to the hosts, after which Messi claimed Brazil were "managing a lot in CONMEBOL these days.".
And when he was harshly dismissed in the next game, which Argentina went on to win 2-1, he couldn't contain his anger.
"Corruption and the referees are preventing people from enjoying the football and they're ruining it a bit," Messi said.
He was given his marching orders after a first-half scuffle with Chile captain Gary Medel in which television pictures suggested he'd done little wrong.