Short answer: Yes, you can sue. No, you won't win.
Long answer: As most other have said, there are numerous warnings on the system startup and in manuals relating to health issues. It's not Nintendo's responsibility to make sure a consumer is fit to buy the system before they purchase it. Their job is to take the steps to warn someone who has purchased the system of the potential health risks involved, and the customer then has the responsibility to take care of themselves. If they don't, it's their fault, not Nintendo's.
Say you buy a guitar amplifier. There would be warnings telling you not to play at high volumes. If you play it loud, and start to notice hearing problems, and then continue to play it too loud, the company that made the amplifier is not responsible for you going deaf, because YOU were the one stupid enough to keep doing something that you were initially told was bad for you.
If playing the Wii hurts you, stop. Sell it. Buy another console, and move on. If you think you're going to get carpal tunnel from playing on the Wii (BTW, typing or any mouse/keyboard usage is far more likely to give you CT anyway) then don't do it. If you don't have enough common sense to stop doing something that hurts you, then you deserve the pain associated with it. Just don't expect Nintendo to foot the bill for it, because they've already done a fine job of covering their asses in this regard with warnings on pretty much anything related to their system. Suing would basically just costs you a bunch of money in lawyer fees and net you nothing in return. Best of luck with the suing, though.
The dedication you show to any particular console or company is inversely proportional to the number of times you have gotten laid. If you get laid enough, even if you prefer a certain brand, you just don't give enough of a shit to argue about it on the internet.