Apple Watch

No.

The Cellular Apple Watch shares the iPhone's phone number.  This requires the iPhone cellular carrier to support the Apple Watch.  The iPhone carrier charges a monthly fee for this service.  In the U.S. the fee is $10 per month added to your iPhone bill.

iPhone Cellular Carriers that support the Apple Watch, which models, the radio frequencies each model uses, and the country where they offer service:

https://www.apple.com/watch/cellular

Your carrier Must be listed by name in the above link. NVMO's that piggy back on another carriers network, unless explicitly mentioned, do not support the Apple Watch.

As fieryjack said, if you have a GPS only model, or do not sign-up for cellular, then the Apple Watch uses the radios in the paired iPhone.

The watch pairs with the phone and uses the phone to make calls and send texts as long as you have the phone with you.So yes it will.If it's the cellular version of the watch then you must use a carrier that supports it and pay an extra fee on top of your existing contract.