The old wrist watches, that had LED displays, had a PB switch to turn-on the LEDs only when you wanted to see the time because of the high power consumption of the LEDs. Some later units used an inertia switch, so you just had to flick your wrist to momentarily turn on the display. Even then the battery life tended to be short.
The wrist watch ICs are a single IC with all the circuitry to do wrist watch functions, such as crystal oscillator, time, day, date, stopwatch, alarm, display driver, etc. Those ICs are all CMOS digital with no analog.
You could conceivably design a wristwatch with a small, very low power microprocessor, but it's rather reinventing the wheel.