The Lampre-Merida rider, who has three stage wins to his name in his career, was withdrawn prior to stage 16 due to a bout of pneumonia.
Costa had been sitting 13th in the standings, 12 minutes and 57 seconds behind leader Vincenzo Nibali of Astana.
The Portuguese is the 29th rider to withdraw from the race, following the likes of defending champion Chris Froome and two-time winner Alberto Contador in retiring.
His exit comes after that of Simon Yates, whose debut on the Tour came to an end on Monday following an impressive showing that saw him finish 15 stages and take part in two breakaways despite suffering from illness of much of the race.
Yates' withdrawal leaves Team Sky's Geraint Thomas as the only British rider remaining in the Tour.