Steven Payne, 37, was attacked at a remote beach about noon on Monday at Wharton Beach, east of Esperance in Western Australia.
A large sea and coastal search failed to find his remains and WA police on Wednesday said it had been suspended "pending any new information being received".
Drone footage from a witness of the attack showed a shark swimming away from a large cloud of blood close to the shore and a surfboard with bite marks was recovered from the water.
Police previously said the search was a recovery operation, not a rescue.
Mr Payne, his partner and dog were five weeks into a six-month caravan trip around the western half of Australia when the incident happened, according to his family.
"Steve was one of the best. A gentle giant at 6ft4 (193cm), he was smart, kind, funny, laid back and very practical," the New Zealand-born surfer's family said in a statement.
"While he wasn't a man of many words, he loved his partner deeply, cherished his family, thought the world of his best mates, and was devoted to his dog."
The presumed death is the fourth fatal shark attack since 2017 in the area, about 800km south of Perth.
The family said Mr Payne, who lived in Geelong, Victoria, would not have supported a shark cull or any attempt to kill the shark responsible.
"He and his partner spent a lot of time in the ocean together, had enjoyed many dives with sharks and knew the risks."
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