A statement from the British Sub Aqua Club confirmed the 55-year-old from Huddersfield went missing during a dive on July 28.
Search operations are being carried out by the coastguard, a specialist police underwater dive team and local dive and fishing boat operators.
However, the conditions are hampering the underwater searches, they said.
The statement said: "Rob commenced as a qualified scuba diver in 1996, he was a member too of the Cave Diving Group.
"He was extremely experienced and well respected throughout the cave and scuba diving communities, having logged in excess of 1,500 dives, all over the world with Melanie, his wife and scuba diving buddy.
"Melanie Dalby wishes to thank everyone for their professionalism and support during this most difficult time.
"Rob worked for Kirklees Council as their council’s green space operational manager, and had worked for Kirklees for 23 years. He was well-respected and liked throughout the authority."
With no police on the Isles of Scilly, Devon and Cornwall Police's diving team have been carrying out searches. The force has been contacted for comment.