After the fifth time trying to update to watch OS 4.3, I was unable to even begin the process again, because it would not pair. A message on the watch said to move the watch next to the the iPhone and a message on the phone said to make sure Bluetooth was on in the phone. The watch was next to the phone and turning Bluetooth on and off could not get rid of that message on the phone. So it was impossible to do a reset and erase from either the phone or the iPhone Watch app.
I was about to call Apple even though I did not have AppleCare on the Watch or iPhone and ask them to fix it since it worked fine until Apple's 4.3 update made it unusable. But then it occurred to me that maybe the problem was with the iPhone Watch app and not the watch or the update. I deleted the Watch app on the iPhone and reinstalled it from the App Store. That did the trick.
Because I had spent more than 5 hours on this problem already, I chose the options I thought might be more likely to work. I do not know if it made any difference, but I chose to set it up as a new watch rather than to restore from the last backup. I also chose to not install all apps, but rather what Apple, I guess deems are the minimum number.
This Apple Watch is the very first edition bought within a month after Apple Watch debued and I installed every update Apple released with nary a problem until this time.