"I have to leave, it's the most honest thing to do," Aguirre told a news conference in Mexico City.
Aguirre had set a target of reaching the quarter-finals but the Argentina defeat condemned Mexico, who last reached the last eight on home soil in 1986, to a second round exit for the fifth finals in succession.
Aguirre said he took "full responsibility" for Mexico's exit.
"The plan was to qualify to the quarter-finals, be among the eight best (teams) in the world. Since this didn't happen, the the team and I agreed that continuing was impossible," Aguirre said.
Mexican media said Aguirre's most likely successor would be Mexican trainer Jose Manuel de la Torre, who guided Toluca to their 10th Mexican league title last month.
Aguirre's first stint in charge of the national team came when he averted a qualifying catastrophe to take the side to the 2002 World Cup in Japan and South Korea, an experience which ended in the first knockout round after a loss to the United States.
He took the helm again in April last year after previous coach Sven-Goran Eriksson resigned following the team's qualifying round loss to Honduras. Aguirre helped resuscitate Mexico's stumbling campaign.
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