West was released from police custody by Thursday afternoon. Blair Berk, who was retained as his attorney, said she couldn't comment on the incident.
Clips of the video shot by the celebrity gossip site TMZ shows West and Crowley grabbing the camera from a photographer, who is yelling "Police! Police!" and "Help me!" After they wrest it away, it appears that the manager smashes the camera to the ground, and the rapper follows by slamming down a flash.
Crowley is then seen accosting the TMZ videographer who's shooting the footage. The video camera was broken in the scuffle before security intervened, the Web site reported.
The video starts mid-confrontation, and doesn't show how it began or what might have caused it to escalate.
Los Angeles International Airport spokesman Marshall Lowe said West and Crowley were scheduled to board a flight to Honolulu, Hawaii on Thursday morning. They had not yet reached a security screening checkpoint when the altercation began.
The airport is a popular spot for the paparazzi, who frequently catch shots of Britney Spears and other celebrities arriving and leaving Los Angeles.
Charles Davis, who solicits charity donations at the airport, said he witnessed the incident, and that West was "attacked" by paparazzi, and the rapper "just defended himself."
"I talk to him all the time," Davis said. "He's a very nice gentleman. Very nice. He gives good donations to help children. He's got a good, kind heart. I've gotten his autograph several times. I just don't see why he was arrested. The man just put the camera too close to his face. I don't blame him."
Davis said he didn't think the camera was destroyed.
"It was just knocked out of the man's hand," he said.
Lowe said police at the airport interviewed witnesses and that West and Crowley were then sent to a nearby Los Angeles Police Department station for booking. West left that police station in a black Chevrolet Suburban by midafternoon, shielded by umbrellas so that photographers couldn't get a shot. He did not give a statement to reporters.
West, who rocketed to stardom in 2004 with his album "The College Dropout" and follow-ups "Late Registration" and "Graduation," has become well-known for his temper tantrums, skewering everyone from MTV to President Bush while the cameras rolled.
But before the Thursday incident at LAX's Terminal 4, his many blowups had always been verbal - making headlines rather than police blotter.
West appeared at the MTV Video Music Awards on Sunday night, which was held at the Paramount Studios lot in Hollywood. His performance of a new song concluded the show and represented a reconciliation with the network after a high-profile tiff last year.
After not winning any awards at the VMAs in 2007 despite several nominations, West declared to reporters that he would never return to MTV. The year before, he crashed the stage at the MTV Europe Music Awards and unleashed an expletive-filed tirade after his video "Touch The Sky" failed to win an award.
In 2005, West famously chastised President George W. Bush during a national telethon to raise funds after Hurricane Katrina struck by saying, "George Bush doesn't care about black people."