The West Ham United midfielder sustained the injury in Mexico's tournament opener against Jamaica on Saturday.
Alvarez fell to the ground clutching his left hamstring and was subsequently replaced in the 30th minute by Monterrey's Luis Romo.
El Tri went on to claim a narrow 1-0 victory, but their successful outing was ultimately soured by Alvarez's early withdrawal.
Alvarez confirms tournament-ending injury
Mexico's worst fears have now been realised after Alvarez confirmed that he will miss the rest of the tournament in a video posted to the national team's social media accounts.
"Unfortunately, my participation in Copa America has come to an end," Alvarez said. "I have no words. The truth is that it is a hard blow for me because I had that dream as much as my team-mates, but sometimes these things happen in football and I will have to learn and grow from all this.
"But I wanted to let you know that I have made the decision to stay with the team until the end, to support and encourage them, as I always have done. Now it will be my turn off the field. It will also help my rehabilitation and return as soon as possible."
While Alvarez will remain in the USA to support his teammates, he will have to hand the captain's armband over to Almeria centre-back Cesar Montes.
Injured Alvarez set to leave signicant void in Mexico side
Speaking before Alvarez's absence was confirmed, Lozano admitted that it would be a "heavy blow" to lose the 26-year-old for the remainder of the tournament.
"I think he's leaving the hotel now to do the examination," Lozano told reporters ahead of Wednesday's game against Venezuela.
"He's expressed that he wants to stay with the group, that he wants to stay here. We'll see what happens, depending on what the examination shows."
Having made 80 appearances for his country, Alvarez's absence will certainly be felt by his teammates, who will now have to raise their game to fill the void left by the West Ham man.
Romo is likely to be given the tough task of replacing Alvarez in the starting lineup for the game against Venezuela at SoFI Stadium in Inglewood.
Mexico could progress to the quarter-finals with a win over Venezuela, depending on the result of the group's other game between Ecuador and Jamaica.