“For me the national team is over. I’ve done all I can, it hurts not to be a champion,” the 29-year-old said.
With Barcelona, Messi has won eight La Liga titles and four Champions Leagues.
But his only major international honour is 2008 Olympic gold, with Argentina now having lost three major finals in three years.
Argentina were beaten 1-0 in the 2014 World Cup final by Germany before two Copa America final defeats by Chile on penalties.
He was also on the losing side against Brazil in the 2007 Copa America final. After Sunday’s match finished 0-0 in 120 minutes, Chile won 4-2 on penalties.
Messi’s penalty miss was Argentina’s first attempt in the shootout and it ballooned over the bar after Sergio Romero had denied Chile’s Arturo Vidal.
Lucas Biglia also missed from 12 yards, with Chilean substitute Francisco Silva scoring the decisive spot-kick in the final in East Rutherford, New Jersey.