Argentina lost to Chile on penalties in the final of the competition, the second time they have lost in the final in the space of two years.
The disappointment of that defeat was compounded by the sobering knowledge that they have not won a major international title for 23 years despite coming close several times in recent years.
Martino's departure comes in the wake of that of his captain Lionel Messi who announced his retirement shortly after the final in which he missed a penalty in the shootout.
The 53-year-old coach took over as Argentina coach in August 2014 and was also due to lead the country in next month's Olympic Games.
However, with the likelihood of the cash-strapped Argentina side competing at Rio looking more tenuous every day, Martino has decided to jump ship.
"The national team's technical team have decided to hand in their resignation," an Argentina Football association statement confirmed.
"Due to uncertainty over the designation of new authorities in the Argentine Football Association and the serious problems in putting together the team that will represent the country in the Olympic Games, the team has chosen to resign."