My Apple Watch won’t pair because of an update and it won’t update

Do you by any chance have an iPhone 6 or older?

iPhone -> Settings -> General -> About -> Model Name

The current shipping Apple Watch Series 3 and 5 models require an iPhone 6s, SE, 7 or newer running iOS 13.  It will Not pair with the iPhone 6 or older.

If this is the case, then return the Apple Watch. Check with your retailer for their return policy. The Apple return policy is 14 days

https://www.apple.com/shop/help/returns_refund

Make sure your phone is fully updated then;

Set up and pair Apple Watch with iPhone - Apple Support

New Apple Watch Series 3 so it should already have watchOS 6 on it. The only way it would not have watchOS 6 was if you purchased it from a retailer that was selling old inventory from before Fall of 2019. Because from that point forward Apple only shipped Apple Watches with watchOS 6.

Now it is possible your iPhone still has iOS 12 on it. If that is the case, then you would need to upgrade your iPhone to iOS 13

Do you know what iOS version you iPhone has?

Finally, if you get stuck, then I suggest making a Genius Bar appointment at your local Apple Store.