Can’t send text messages on Apple Watch 3 when iPhone is not near.

Welcome to the Apple Support Communities!

If you are not able to send text messages from your Apple Watch Series 3, take a look at the troubleshooting steps outlined in the resource below:
Read and reply to messages with your Apple Watch

Get help if you can't send or receive messages

After each step, try to send or receive a message again:

1. See if you can send and receive messages on your iPhone.
2. Make sure that your Apple Watch can connect to your iPhone or Wi-Fi.
3. If you have Apple Watch Series 3 (GPS + Cellular) and it's connected to a cellular network, you can check the signal strength in Control Center. To send and receive SMS or MMS, your paired iPhone must be powered on and connected to Wi-Fi or cellular, but it doesn't need to be nearby.
4. Make sure that you signed in with your Apple ID on your iPhone and Apple Watch. On your iPhone, open the Apple Watch app, tap the My Watch tab, then tap General > Apple ID.
5. If you see on your watch, turn off Do Not Disturb.
6. Check the person's phone number or Apple ID to make sure that it's correct.
7. Sign out of iMessage on your iPhone. If you're signed in with your Apple ID, tap it, then tap Sign Out.
8. Turn iMessage off on your iPhone, then turn it back on again.
9. Sign back in to iMessage on your iPhone.
10. Restart your Apple Watch and iPhone.
11. Unpair your Apple Watch, then pair it again.

Take care.