Is COX internet required to use apps like Netflix through the contour box?

Although your Contour box is connected to the TV distribution network (separate from Internet), you will need Internet service with Cox to use an app on the Contour box.

Streaming programs is from an Internet source.  Cox measures your Internet data use.  Although your request to stream from Contour is over the TV distribution network, Cox needs to measure your Internet data use when you stream on an app with Contour.  Cox can't measure your use if you don't have an Internet account, so the app on Contour wouldn't connect.