The 24-year-old was suffering from what appeared to be a knee injury that flared up in the second set, and he eventually pulled out at 3-6, 6-3, 4-1 down against the Argentine.
Draper returned from a six-month absence in February after suffering a left arm injury which forced him to withdraw from the US Open in August 2025.
This was his first match of the clay court season, having pulled out of the Monte Carlo Masters, and his fourth event since returning from the injury in his serving arm.
The Englishman is currently ranked 28th in the world but risks dropping significantly if he cannot compete at the Madrid Open next week.
Last year Draper finished runner-up to Casper Ruud in the Spanish capital, leaving him with points to defend.
World number two Carlos Alcaraz is the top seed in Barcelona and takes on Finland's Otto Virtanen on Tuesday.
