Can I pair my Apple Watch to a new iPhone without backup?

Try this

If you can't unpair Apple Watch because you don't have its paired iPhone, you can erase the watch.

When you erase Apple Watch instead of unpairing it, the watch is still protected by Activation Lock. Before you can pair Apple Watch again, enter the Apple Account and password that were used to set it up.

1. Open the Settings app on your watch, then tap General > Reset.
2. Tap Erase All Content and Settings, then tap again to confirm.
3. Wait for the watch to unpair.
4. After Apple Watch unpairs completely, you can pair again. You'll need the Apple Account and password that were used to set up the watch.

To answer your implied question, pairing without the backup mostly loses your saved faces, not much more