No, properly used watch winders do not damage watches. In fact, they can be beneficial for maintaining and prolonging the life of automatic watches. A watch winder keeps the movement running and lubricated, preventing the oils from drying up and potentially causing damage, especially if the watch isn't worn regularly.

Here's why watch winders are generally safe and even helpful:
Overwinding Prevention:
Modern automatic watch movements are designed with a clutch mechanism that prevents overwinding. Even if the winder rotates the watch continuously, the movement will simply dump any excess energy, preventing damage.
Lubrication Maintenance:
Keeping the movement running evenly helps maintain the lubrication of the internal components, which can degrade if the watch is left still for extended periods.
Complication Management:
Watch winders are particularly useful for watches with complications like perpetual calendars or moon phases, as they ensure the watch stays at the correct time and date without manual adjustment.
Convenience:
Watch winders make it easy to keep a watch running and ready to wear, especially if you have a collection of watches that you rotate.

However, there are a couple of points to consider:
Low-Quality Winders:
Poorly made winders or those that don't match the specific requirements of your watch (e.g., rotation direction or turns per day) could potentially cause issues, such as magnetization or excessive wear on certain parts.
Constant Rotation:
While not inherently harmful, continuous rotation in a winder can potentially lead to slightly accelerated wear and tear on the movement. However, this is generally minor and outweighed by the benefits of consistent winding and lubrication.
In conclusion, a well-chosen and properly used automatic watch winder box is a valuable tool for maintaining the health and longevity of your automatic watches.

This video explains whether watch winders are bad for your watches: