Australian Broadcasting reports Irwin's wife, Terri, has turned down the offer of a state funeral.
In the first public comments by Irwin's family since the hugely popular naturalist died Monday, Bob Irwin thanked his son's many fans for their messages of support, and said his son had died doing what he loved.
The 44-year-old TV star was on camera for a new program as he snorkeled with a stingray on the Great Barrier Reef, when it lashed out with its tail, plunging a poisonous barb into his chest. Irwin died minutes later.
Separately Wednesday, Irwin's manager and close friend John Stainton said videotape showing him being fatally stabbed should never be publicly aired.
"It should be destroyed," Stainton, who has seen the footage and said it showed Irwin pulling the barb from his chest in his last moments, told CNN's "Larry King Live."
The tape is now in the possession of police as evidence for a coroner.
"When that is finally released, it will never see the light of day. Ever. Ever. I actually saw it, but I don't want to see it again," Stainton said.
The Discovery Channel, which produced and aired Irwin's programs to a reported global audience of more than 200 million, said it will not show the footage.
Thousands of fans have flocked to Irwin's wildlife park in Australia's Queensland state, Australia Zoo, creating a shrine of flowers and written tributes. Prime Minister John Howard said a state funeral was appropriate, calling him a great ambassador for Australia.
Bob Irwin said a state funeral wouldn't be what Steve wanted.
"He's just an ordinary guy, and he wants to be remembered as an ordinary bloke," said Bob Irwin. "The state funeral would be refused."
Bob Irwin, who started the wildlife park that his son turned into a major tourist attraction, said Steve realized his work was dangerous and that he could die doing it.
"Steve knew the risks involved in the type of work he was doing, and he wouldn't have wanted it any other way," Bob Irwin said.
Michael Hornby, the head of one of Irwin's wildlife charities, Wildlife Warriors, said Terri Irwin is considering having a smaller, private ceremony at an Outback location, and approving a separate large event at a stadium in the state capital, Brisbane.
Hornby said Wednesday that two or three bogus Web sites had been set up attempting to divert some of the money being donated to Irwin's charities following his death.
"There are leeches around the planet, and we are fearful that more are going to come up," said Hornby. "We want people to make sure they donate only to wildlifewarriors.org.au or through the U.S. Discovery Channel, the Crikey fund."
Police have said there are no suspicious circumstances in Irwin's death, and no decision has been made about whether a coroner will hold a formal inquest or simply accept the police findings. No formal cause of death has been announced.
Inquests are held into suspicious deaths, if relatives ask, or if the coroner decides the public interest would be served by one.
"Over the years Steve and I had a lot of adventures together," said Bob Irwin, 66, sitting before a throng of news cameras, journalists and well wishers at the gate of Australia Zoo.
"Steve and I weren't like father and son, we never were. We were good mates. I will remember Steve as my best mate ever."
Terri Irwin briefly addressed park staff late Tuesday over a public address system, thanking them for their support in "very choked up ... very frail" comments that were her first since Irwin's death, Hornby said.
Bob Irwin was in a similar frame of mind Wednesday, admitting before his media appearance, "I'm not sure whether I'm going to be able to get through this."
He ended up doing fine, and asked for the media in turn to respect the privacy of Terri, originally from Eugene, Oregon, and their two children, Bindi, 8, and Bob, 2.
Bob Irwin had just spent nearly a month with the entire family on Cape York in tropical northern Australia doing crocodile research.
"Steve was probably the best I had seen him in many years, in his own personal attitude," he said. "He was peaceful. He was not under stress. And he was doing something that he really loved doing. I won't ever forget that three or four weeks."