The 27-year-old had been undergoing medical treatment since collapsing during Nacional's game against Brazilian side Sao Paulo on August 22.
Izquierdo fell to the ground unconscious in the 84th minute without coming into contact with another player. He was taken off the pitch in an ambulance before being treated at the intensive care unit of Albert Einstein Hospital.
The hospital said in a statement Izquierdo had died as a result of brain death following a cardiopulmonary arrest associated with cardiac arrhythmia.
"It is with the deepest sorrow and shock in our hearts that the Club Nacional announces the death of our beloved player Juan Izquierdo," Uruguay's Nacional wrote on X.
"We express our most sincere condolences to his family, friends, colleagues and loved ones. All of Nacional is in mourning for his irreparable loss."
The Uruguayan Football Association said the news was deeply painful, while the national governing bodies of Argentina, Peru, Paraguay and Colombia extended their condolences to Izquierdo's family and friends.
"South American football is in mourning," said Alejandro Dominguez, the president of South American soccer's governing body CONMEBOL.
Brazilian Football Federation President Ednaldo Rodrigues said a minute's silence would be observed in every match organised by the federation on Wednesday.
Sao Paulo said they were deeply saddened by the news, calling it a "sad day for football".
