We have the same problem we had with the Series 3 watch...I am unable to make or receive calls when the phone is off. We got a GPS only Series 4 for DH as he never once used the cell capability on his watch. He can't receive texts at all. After 1+ hours on the phone with AT&t and a good .5 hour with Apple, we are headed to the Genius Bar later this afternoon. I'll be really pi$$ed if the Genius again tells us that the phone has to be ON to make/receive calls.
After an hour with the geniuses at the Genius Bar, we are told the Apple Watch CANNOT make or receive phone calls when the iphone is OFF. At first about half said yes and half no. I told them to fix my phone. At the end, all of the geniuses said the watch will not work when the phone is off. One even showed me 'proof' on their website...."To receive SMS, MMS, or push notifications for third-party apps on your Apple Watch with cellular, your paired iPhone must be powered on and connected to Wi-Fi or cellular." I questioned what this has to do with sending and receiving calls and she said this covers it.
Anyone out there who can disprove this? Can you do some testing with your watch and phone? This is driving me nuts.....