This happened to me, and support didn't really help. Basically, the app was actually updated but the Galaxy Wear app on my phone didn't think it was.
Here is how I fixed it:
1. I had to factory reset
2. Paired to phone, and signed into Samsung
3. Immediately turned auto update off in the phone app settings
4. Turned Bluetooth on my watch off to disconnect my phone
5. Connect to wifi on the watch (note: it seems like only 2.4ghz signals work)
6. Update the app directly on the watch (home button > Galaxy Store > tap three dots on the side > My apps > Samsung Health > Update)
7. Turn Bluetooth back on and it worked! Finally!!!!
The problem is not only with Samsung health, but also with Samsun Internet browser. Please find a way to fix it, because it is really annoying not being able to update your apps.