The Colombian, who was set to be among the Quick-Step Floors leaders for Saturday's Milan-San Remo, will require surgery to fix the break, necessitating four weeks off the bike and so he will miss the Monument.
He was also due to ride the cobbled Classics, including debuts at the Tour of Flanders (April 1) and Paris-Roubaix (April 8), but that now seems unlikely.
Gaviria was riding behind his Quick-Step Floors train with 8km to go in the stage to Fano when he touched wheels with a teammate and crashed heavily.
The 23-year-old remounted and finished the stage, his shorts ripped open on the left side. X-rays at the finish line revealed a broken metacarpal, one of the bones on the back of the hand.
"He’s got a displaced fracture of his first metacarpal of his left hand," the orthopaedic doctor told the media at the x-ray vehicle.
"One part of the bone has moved over another part. It needs to be put back in place. He’ll need surgery to reduce the fracture and fix it in in place.
"To make a full recovery he’ll need at least four weeks. It’ll be difficult for him to ride before that, the fracture needs time to stabilize."
Gaviria's crash is his second of the year. He won the opening stage of the Vuelta a San Juan, but crashed out of the race on stage 4, suffering a deep gash to his knee.
In 2017, Gaviria won four stages of the Giro d'Italia and the points classification, but then suffered a contusion to his leg that required surgery in July. He rebounded to win six more times before the end of the season.