No, the small device supplied with the watch has more available power than the watch will request. The charging current is actually controlled by the device being charged, subject to the maximum the adapter can supply. If the adapter has a higher rating the excess capability is simply not used - think of it like driving a 500hp sports car in city traffic, where your right foot never asks for more than 50 hp.
I believe Apple’s smallest charger power adapter is rated at 5W (1000 mA), double the basic USB port on a computer which can output 500 mA. Your Apple Watch probably reaches maximum charge rate around 200-300 mA (my guess based on reported battery size) so the standard charger will never reach maximum output. The watch software will not allow it to charge faster or at a potentially unsafe rate.