No, a watch winder can’t damage your watch. Your watch movement requires a minimum amount of tpd (turns per day) to keep it ticking away. Should you exceed this, then the movement has an inbuilt clutch type mechanism that simply dumps any excess energy created with zero impact or detriment to your watch. For this reason, you cannot overwind an automatic watch.
To illustrate this, you might ask yourself ‘how many tpd do I generate for my watch each day?’ The question is pretty much unanswerable. In reality we have no idea how much energy we generate each day for our watch, it will depend on how active a lifestyle you lead. For instance, you may have quieter days and more lively days, but it would never cross your mind to remove the watch for fear of overwinding it!
For example, casting your mind back to that wonderful day in the jewellery shop when you decided to actually buy the watch, the store assistant will have sat in front of you and given the watch a full manual wind using the winding crown, and then you put it on your wrist and continued to wind it kinetically for the rest of the day. At no point did you consider taking it off for fear of overwinding! The same with a watch winder – you cannot over wind an automatic watch.