Can I use my Call of Duty purchase from Battle.net on Steam

Cross-Progression allows you to play a game on PC, console, or mobile while keeping progress and collections across all devices. For paid games, Cross-Progression does not mean that you can buy a game for one platform and then play it on any platform, you are still required to buy the game for each platform. 

Call of Duty game progression is stored on your Activision account. Your progression (stats, rank, weapon, and camo progression) and unlocked content (camos and other customization items) will be available on all platforms as long as you play on the same linked Activision account where you originally earned the progress and unlocked the content.