How do I change the background color of my watch face? (not the color of the text or digits)

You can change clock faces … and the specifics of the clock face … as the specific clock face allows. To change the background color, you have to select a clock face that allows you to do that. Are you asking about a specific clock face?

California allows you to change the background color of the clock face. So does Color. There may be others.

Change the watch face

Explore the Face Gallery on Apple Watch

Which watch face? Not all allow you to change the background color.