The problem with that sentence is that the subject that you'd be watching is not stated, so the only other subject that "watch" could refer to is the glasses. This would imply that you need to watch you glasses while you're reading, which might make it difficult to keep your attention on the book!
The sentance grammatically is fine, it just doesn't make much sense.
"I need a television to watch when I'm ironing" is fine, because the thing that you're watching is the television. But why would you want to spend the whole time you're supposed to be reading watching your glasses - their not going to go anywhere!
So in answer, either sentence is grammatically correct, but if you mean that you wear glasses when you read, "wear" is correct.
I need glasses to ___ when I'm reading.
I agree with yours and FRC's and Sam's choice. :D 'wear' is the correct choice. 'watch' is not even an option. It's a transitive verb, which means it needs an object, like this,
I need glasses to watch a movie when I'm at the cinema.
Also, 'watch' and 'reading' are not compatible. The verb watch goes with movies, TV, animals, babies, tennis matches, and anything else that moves. To watch something means to follow its movements. A book, sadly enough, doesn't move, nor does its pages. :wink: