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.

The sub was 20km (12 miles) from the Admiral Kuznetsov at the time of the incident, the military said.
Russia sent a flotilla to the eastern Mediterranean ahead of an expected resumption of air strikes in Syria.
Nato said it was monitoring the ships in a "measured and responsible way".
An official said the Western military alliance would not go into details but said it had been observing the fleet for some weeks.
The Russian flotilla's conspicuous voyage from Severomorsk in northern Russia, through the North Sea and the English Channel and into the Mediterranean, has rung alarm bells among Nato allies.
There was no confirmation of the Russian military's claim of an incident, and the Dutch military tweeted that it would give no comment on submarine operations.
It was also not immediately clear where the incident took place. However, one report placed the Admiral Kuznetsov around 100km (62 miles) north-west of the Syrian port of Latakia.
Russian defence spokesman Igor Konashenkov said that two anti-submarine ships, Severomorsk and Vice-Admiral Kulakov, had "easily identified the submarine 20km away using the standard onboard hydro-acoustics systems and data obtained from anti-submarine helicopters".
Although the sub had tried to evade surveillance, the ships tracked it for more than an hour and "forced it to leave the area of the aircraft carrier-led group", he said, condemning the Dutch sub's manoeuvres as "clumsy and dangerous".
Although there was no official reaction from the Dutch military, Defence Minister Jeanine Hennis-Plasschaert said she thought it wise "not to go along with the Russian rhetoric".
Dutch marine expert Jaime Karreman said that if true the incident was unprecedented. "It's pretty shocking that a submarine on a secret mission is discovered," he told Dutch TV.
Enrique Iglesias's hand injuries are worse than expected, his spokesperson has said.
[NEXT_CONCEPT]
Russia's defence ministry says two of its navy destroyers forced away a Dutch submarine to stop it spying on an aircraft carrier in the Mediterranean.