Dawn Brancheau, 40, died Feb. 24 when the whale named Tilikum grabbed her by ponytail and thrashed her around after a show. The whale has also been involved in two other deaths, one of a trainer at a Canadian park and another of a man who snuck past security and was found draped over the whale's body at SeaWorld.
Jon Mills, an attorney for Brancheau's family members, said in court that their right to privacy outweighs the public's right to view the video captured by SeaWorld cameras. At a hearing that lasted less than hour, he asked the judge to permanently stop the video from being released.
"There is no constitutional right to voyeurism and there is a constitutional right to privacy," Mills said.
Circuit Judge Lawrence Kirkwood did not immediately rule on the request for the permanent injunction. He had issued a temporary injunction last week to prevent the release of the video, which was turned over to law enforcement.
Unless Kirkwood intervenes, the material will become public under Florida law once the Orange County Sheriff's Office concludes its investigation.
Rachel Fugate, an attorney representing news organizations, said she didn't object to an extension of the injunction so all sides could reach a resolution.
She said there were precedents in recent Florida history that allowed news organizations to view images but not make copies for public dissemination. Members of the public, for example, were allowed to view photos of the mutilated bodies of five college students slain in Gainesville in 1990, but not copy them.
Grieving relatives have succeeded in keeping such images from being made public in other cases - Dale Earnhardt's widow successfully fought in Florida to prevent the release of autopsy photos of the race car driver, who died in a 2001 crash.
Earlier this month, officials released recordings of the 911 calls made after a Brancheau was pulled into the water by the whale.
Investigators have said Brancheau died from multiple traumatic injuries and drowning. The 22-foot-long whale pulled Brancheau into the pool by her ponytail last week after a performance.
In police investigative reports released earlier, Brancheau's co-workersafter alarms sounded around the pool. Employees who were at other pools, behind computers or emptying coolers of fish rushed to the scene.
More coverage of the SeaWorld tragedy
Worker: "Whale Would Not Let Us Have Her"
SeaWorld Trainer Mourned in Chicago
Whale Attack Renews Captive Animal Debate
Tears Flow as Whale Shows Resume
SeaWorld Called Best Place for Tilikum
Whale, Trainer Play on Film Before Attack
SeaWorld Defends Serial Killer Whale
SeaWorld: The Show Must Go On
Ex-SeaWorld Official: Trainer Made Mistake
Dawn Brancheau's Last Moments Alive
SeaWorld Trainer Embraced Risks of Job
Mills spoke to "Early Show" co-anchor Harry Smith about the video: