Back-row star Corry, who led England in two of their RBS 6 Nations games last season, has also agreed a new long-term deal with Tigers.
"The important thing is that I don't try to copy Martin," said Corry.
"Sure, we both lead by example, but I don't have all his attributes, starting with his technical knowledge and captaincy experience.
"All I can do is be true to myself. As a captain, you want your players to trust you, and they won't trust you if you try and change who you are."
Johnson and his fellow England World Cup winner Neil Back have both retired, leaving Tigers head coach Pat Howard to find a new skipper.
Tigers' New Zealand centre Daryl Gibson, meanwhile, will be 31-year-old Corry's vice-captain.
"Some people think it was an easy decision, but with Martin's England commitments, it was a tough choice to make," said Howard.
"But Martin is a great character. He loves the club and the players, and it's a great challenge for him to replace an icon in world rugby."
Corry started seven of England's eight Tests last term, and featured in all three Lions Tests against New Zealand, and he is relishing the demands of another campaign for club and country.
"I never wanted to go anywhere else, and I was delighted when the club offered me a new contract," Corry added.
"I would love to finish my career here - it's something I have always wanted to do since joining Leicester in 1997."