Do this if your Apple Watch is calling SOS:
1. Open the Watch App on your iPhone
2. Tap Emergency SOS
3. Disable Hold side button to Dial

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Apple Support Forum Post:
https://discussions.apple.com/thread/...

UPDATE: July 01, 2023
After Support chat with Apple, I am taking my Apple Watch to a local authorized repair shop. Support was quick and friendly. I suspect there must be a hardware way to reset the Apple Watch when the software is stuck. Apple logo still appears and disappears over and over while watch is on charger, and there is nothing else I can do right now.

Background: June 20, 2023
For the second time in 2 weeks, my Apple Watch S6 randomly starts signaling SOS, alerting and counting down to call emergency services. Both times, I very sweaty - once after jogging several miles and once after doing yard work.

The alerts don't start when doing anything rigorous, so it is not a false fall detection, I am guessing it is falsely thinking I have stopped moving, as in dead or in trouble, but that is not the case.

In both cases, it seems that somehow sweat is triggering the problem. Almost like the sweat is getting in to the watch case, and making the watch thing both buttons are being pressed and help… or whatever button combo triggers the SOS feature.

I disabled emergency calls in Watch Settings, so something is seriously messing up. I saw there was an update for Apple Watch, but it won't install. I started the installation on the Watch App on iPhone, entered my passcode on the watch, and placed watch on the charger - it's not charging and the Apple logo keeps flashing on and off over and over instead of charging.

In the attached video, you can see everything happening. At about 50 seconds in, the false button presses start to occur.