The 23-year-old, who is the Highland club's first signing this summer, has joined on a one-year contract with the option of a further year.
"It's great to be joining a club who has been successful recently," Wedderburn told the Inverness website.
And manager John Hughes added: "He is a big strong lad with good speed and agility."
Scottish Cup holders Caley Thistle say Hughes had been tracking former Stoke City and Notts County player Wedderburn during 2014-15.
"Nat will be a great addition for the club," said Hughes.
"He is joining a great club at a fantastic time and we look forward to him developing in to a regular first-team player. He is stepping up a few levels, but we know he is well capable of doing so."
Wedderburn expects the step up from Championship to Premiership level to be "a challenge".
"I'm ready to take it," he explained. "I don't expect to go straight into the team. It's like when you go to any new club, you have to earn that right, but I am here to work hard and play my part."
Meanwhile, Inverness chairman Kenny Cameron revealed the club have been unable to secure striker Edward Ofere for next season after "extensive discussions" with the player's agent.
And talks are continuing with long-serving midfielder Nick Ross, who Cameron says has been made "a very good offer".
"It's pleasing to have kept the nucleus of the squad together and we thank those moving on for the part they have played whilst here," added Cameron.
"John has a number of targets in mind and we are confident of adding another three or four faces to the squad in the very near future."
Inverness Caledonian Thistle have signed Nat Wedderburn following the midfielder's exit from Cowdenbeath.