If all work done on the website is owned by your company then unless there is some specific part of the contract he is quoting it sounds like he's out of luck. That said, if you are able to make a backup of any files pertaining to the website, do so ASAP.
You can try threatening him with legal action, in which case:
- Get as much evidence ASAP, before you threaten (emails, signed contracts).
- Backup as much data as you can, in case things don't go well and your webmaster becomes aggressive.
- Present the evidence and tell him that you will take legal action. This is often enough. If he is part of a larger design company, inform the company of the situation and your next steps.
- Hire a lawyer.
I also imagine that this is a common practice, known as extortion.
EDIT - This assumes no other important details. I don't know your contract and I don't know why he feels he may have been wronged. Also I am not a lawyer.