China's government signalled its rising anger at the protesters, denouncing some of the violent demonstrations during the weeks of unrest as "terrorism".
The United States urged "all sides" to refrain from violence and Canada called on Beijing to show restraint.
Following Monday's shutdown, authorities said they sought to reopen the airport at around dawn on Tuesday, but hundreds of protesters had remained in the arrivals hall well into the night.
It was the 10th consecutive weekend that protesters had taken to the streets in a movement that began over opposition to a bill allowing extradition to mainland China.
The protests have morphed into a broader bid to reverse a slide of democratic freedoms in the southern Chinese city, which is a global financial centre.
Early Tuesday, the website of Hong-Kong-based carrier Cathay Pacific showed all its arrivals and departures still cancelled into the afternoon.
The airport listed some long-haul flights of other airlines as having departed overnight while others had landed.
Other flights, within Asia and beyond, are scheduled throughout Tuesday morning but many were still shown as cancelled.
The disruption marked yet another dramatic escalation in a 10-week crisis that has already become the biggest challenge to Chinese rule of Hong Kong since the 1997 British handover.
On Monday a crowd of protesters that authorities said numbered more than 5,000 descended on the airport carrying placards and chanting slogans about police violence.
Although other rallies had been held there over the previous three days, Monday's was the first to significantly affect passengers.
The airport authority said operations "have been seriously disrupted as a result of the public assembly".
More than 250 flights were cancelled into Tuesday at the world's eighth busiest airport, which handles around 200,000 passengers a day.
Cathay Pacific advised passengers to "postpone non-essential travel".
The airline, which has come under pressure from Beijing, also warned its own staff that they could be fired for supporting "illegal protests".
At the airport, protesters held signs reading "Hong Kong is not safe" and "Shame on police".
They accuse the force of using increasingly violent and disproportionate tactics to suppress protests.
Over the weekend police fired tear gas into subway stations and crowded shopping streets in confrontations with protesters at nearly a dozen locations across the city.
Protesters responded by hurling bricks and spraying riot police with fire extinguishers and water hoses.
Demonstrators were also enraged at police apparently dressing in the black T-shirts worn by the pro-democracy movement to infiltrate the rallies and make surprise arrests.
Officials said 45 people were hurt in the clashes.
Among them was a woman who was reportedly hit by a bean bag round.
Images of her lying on the ground with blood pouring from her face went viral and featured on posters calling for new demonstrations, including some reading "an eye for an eye".
The expression was also spray-painted in Chinese at several places in the airport, where many demonstrators wore eye patches or bandages in solidarity with the wounded woman.
"It is becoming more and more dangerous, but if we don't still come out at this point, our future will become more frightening, and we will lose our freedoms," said one 22-year-old protester who gave her family name as Chan.
The city's transport chief Frank Chan said Monday that Hong Kong would pay a "heavy price" for the airport's closure.
In Beijing, authorities slammed violent protesters who threw petrol bombs at police officers, linking them to "terrorism".
"Hong Kong's radical demonstrators have repeatedly used extremely dangerous tools to attack police officers, which already constitutes a serious violent crime, and also shows the first signs of terrorism emerging," said Yang Guang, spokesman for the Hong Kong and Macao Affairs Office of the State Council.
"This wantonly tramples on Hong Kong's rule of law and social order."
Later Monday, two state media outlets ran videos showing armoured personnel and troop carriers purportedly driving to Shenzhen, which borders Hong Kong.
A commentary early Tuesday in China's official Xinhua news agency spoke of "black-clad mobsters" and said Hong Kong's future is at a "critical juncture".
A senior official in the administration of US President Donald Trump on Monday urged "all sides" to avoid violence in Hong Kong.
"Societies are best served when diverse political views are respected and can be freely and peacefully expressed," the official said on condition of anonymity.
Canadian Prime Minister Justin Trudeau called on China "to be very careful and very respectful in how it deals with people who have legitimate concerns in Hong Kong".