Well, I tried the steps listed below and the one about stopping and restarting iMessage seemed to do the trick. After I did it the first time, I got that there was an error activating iMessage. So when I tried again, I got a message that iMessage was now registered (I forget if this was the exact word) with a new Apple Watch. I retried sending the SMS message and it went right through. So I posting this in case anyone else has this problem: Here are the steps:
It may help to follow these steps:
On your iPhone, go to: Settings > Messages > Send & Receive:
Check that your iPhone number is selected for "Start New Conversations From".
On your iPhone, go to: Settings > Messages:
Check that Send as SMS is enabled.
It may help to toggle the setting off and on again.
Turn off iMessage at the top of the screen, pause for a few moments and then re-enable it.
Check that your Apple Watch and iPhone are connected:
(This took two tries but it worked for me)
About Bluetooth and Wi-Fi on Apple Watch - Apple Support