My new Apple Watch series 3 is not pairing with my iPhone 6

If you just purchased the Watch, it most like has watchOS 6 loaded on it. It requires a phone with iOS 13 to pair to. That means an iPhone 6s or later.

The new Watch Series 3 runs on watchOS 6.x which can only pair with

an iPhone 6s or later models running iOS 13.x or later.

You have 2 choices:

1. Return the Watch Series 3
2. Or buy a newer iPhone to pair with your Watch.

Apple Holiday Return Policy (Purchase between: 15-Nov-2019 thru 25-Dec-2019, Return by 8-Jan-2020)

All purchases made before November 15, 2019 and after December 25, 2019, are subject to the Standard 14-day Return Policy. 

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