Q. Can one wear a kinetic watch on Shabbath, that is similar to a self-winding which poskim permit when it is already working, but in this type the hand movement instead of winding a mechanical device, charges a battery that operates the watch?

A. On question 300 we wrote: “Many Poiskim permit the use of self-winding watches, while they are still working (see Shemiras Shabbos K’hilchoso 28:28). A watch that gets charged automatically from the light it receives and is ticking would also be permitted. However Horav Shlomo Miller’s Shlit”a opinion in regards to the eco-drive watch you describe, is that it should not be used on Shabbos, since your covering or uncovering of the watch will have a direct effect and change the operation and functioning of the watch.”
Eretz Hemda (.org) writes: “The recent invention of kinetic watches is slightly different from self-winding watches. The mechanism of kinetic watches is based on a quartz system, which is normally operated by battery. In this case, instead of a battery, movement generates the small amount of electricity that the watch needs, and the watch stores the energy for anywhere from days to months. The relatively new question of a mini, mechanical electricity recharger is not a simple one, and we have found rabbis coming out in either direction. However, our feeling, is that such a transfer of energy of movement to electricity to be transferred again to “harmless” mechanical movement, without creating new circuits, is not included in the prohibitions of electricity that the halachic world has assumed since the advent of the use of electricity. (We may reverse our decision if a consensus forms to forbid it.)
The matter is more complicated regarding kinetic watches with digital displays, which use electrically generated forms. However, there are significant reasons for leniency here too. The most important one is that the energy provided on Shabbat is rarely needed for operation on Shabbat, as the charge lasts at least a few days.”
Horav Shlomo Miller’s Shlit’a opinion is to be stringent on the use of kinetic watches.
 
Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit’a