Can I charge my Apple watch with an iPhone adaptor?

Depends on the definition of "adapter".

The part you plug into the wall that has a USB input: Yes, you can use that for nearly any USB-supported product for charging.

The cable itself: It has to be an Apple Watch cable or supported device, like a Qi charger which is rumored to work as far as charging the Watch.

http://appleinsider.com/articles/15/04/24/apple-watch-found-to-be-compliant-with -qi-wireless-charging-standard

Yes.