Does Verizon charge extra to add a watch?

$10 per month added to your iPhone cellular bill.

This ASSUMES you are using a Cellular Apple Watch, and that you decide to activate the Apple Watch cellular radio.

If you have a GPS only Apple Watch, Verizon does not know, and does not care, and of course you do not get charged.

if you have a cellular Apple Watch but do not activate the cellular radio, again Verizon does not know, and does not care, and does not charge you anything.

A GPS only (or cellular without activating the radio) works the same as an activated cellular Apple Watch As Long As your iPhone is nearby.

You’ll have to check with Verizon.