The 40-year-old sliced his fingers open when he tried to grab a drone camera, which was taking pictures of the audience, at a concert on Saturday.
Following reconstructive surgery on Monday, it was announced he would need several weeks to recover.
"The damage ended up being a bit worse than initially assessed," said a statement on the star's website.
"He suffered a fracture that needed to be corrected in addition to the reconstructive surgery.
"He will be recovering during the next few weeks, but a full recovery is anticipated."
Iglesias was performing in Tijuana, Mexico when he reached for the drone, apparently in an attempt to give fans a point-of-view shot from the stage.
He was "semi-treated" at the side of the stage to stop the bleeding, and was advised to end the show.
However, he performed for another 30 minutes before seeking further treatment.
"He was rushed to the airport where an ambulance met him," said a representative on Instagram. "He was then put on a plane to LA... to see a specialist."
Afterwards, fans posted photos and video on social media, showing the singer's bandaged hand and blood-covered T-shirt.
They also scribbled get well messages on their own hands and sent pictures to Iglesias via Twitter and Instagram.
Iglesias is one of the highest-selling Latin musicians of all time, shifting more than 137 million albums worldwide.
His UK hits include Bailamos, Hero and I'm A Freak.
The singer is expected to resume his tour on 3 July in Mexico City.
Enrique Iglesias's hand injuries are worse than expected, his spokesperson has said.