There's a geek answer to this, and a practical answer to this.
The geek answer is that there are all sorts of electronics that can have no conceivable effect upon a plane in flight.
The practical answer is that even if you're completely certain that your gizmo can't affect a plane, it's going to be up to the cabin crew to decide whether you are breaking airline regulations. This can cause an effect ranging from being politely asked to turn off the device... to your plane diverting in mid-air so you can be removed from the plane in handcuffs.
So my recommendation for flights in the US: follow the instructions you're given. One flight I was on said, "If your device has an off switch, turn it off." In this case, your watch would be fine. Most flights said, "Turn off your GPS." In this case, I would take out the battery.
It doesn't have to make sense. The better question is, "What are the consequences for disobeying the instruction?" Don't rely on the common sense or tech savvy (or lack thereof) of the cabin crew who are legally empowered (in the US) to make any decision they like if they think it affects the safety of the plane.