The scuffles broke out in the evening between police and protesters, after an injured person was taken out of the main terminal by medics.
Several police vehicles were blocked by protesters, and riot police moved in, pushing some protesters back and using pepper spray at times amid heated scenes.
Protesters also barricaded some passageways in the airport with luggage trolleys and other objects.
Hong Kong’s Airport Authority said operations at the airport had been “seriously disrupted" and that departing passengers had been unable to reach immigration counters.
Hong Kong’s stockmarket fell to a seven-month low.
Ten weeks of increasingly violent clashes between police and protesters have roiled the Asian financial hub as thousands of residents chafe at a perceived erosion of freedoms and autonomy under Chinese rule.
The United Nations High Commissioner for Human Rights urged Hong Kong to exercise restraint and investigate evidence of its forces firing tear gas at protesters in ways banned under international law.
“Take a minute to look at our city, our home," chief executive Carrie Lam said, her voice cracking, at a news conference in the government headquarters complex, which is fortified behind 6-foot (1.8-m) -high water-filled barricades.
“Can we bear to push it into the abyss and see it smashed to pieces?"
China this week condemned some protesters for using dangerous tools to attack police, calling the clashes “sprouts of terrorism". They present President Xi Jinping with one of his biggest challenges since he came to power in 2012.
Hong Kong legal experts say Beijing might be paving the way to use anti-terrorism laws to try to quell the demonstrations. The clashes at the airport followed an unprecedented airport shutdown on Monday.
Again on Tuesday, thousands of black-clad protesters jammed the terminal, chanting, singing and waving banners. Floors and walls were covered with missives penned by activists and other artwork. Initially, the scene was peaceful as knots of protesters spoke to travellers, explaining their aims.
“Sorry for the inconvenience, we are fighting for the future of our home," read one protest banner at the airport.
“I think paralyzing the airport will be effective in forcing Carrie Lam to respond to us ... it can further pressure Hong Kong’s economy," said Dorothy Cheng, 17.The weeks of protests began as opposition to a now-suspended bill that would have allowed suspects to be extradited to mainland China, but have swelled into wider calls for democracy.