From Blizzard Support:
Blizzard Support - How to Install Call of Duty: Modern Warfare III
From Activision Support:
https://support.activision.com/modern-warfare-iii/articles/modern-warfare-iii-installation-and-setup
Step 1: Click on the GAMES button on the Battle.Net Desktop App
Step 2:
- Click on Activision in the left margin
- Click on the star on the Call of Duty icon to add Call of Duty to your Favourite Bar
- Click on the Call of Duty icon
You should now have an install button; click it. If not, give me a screenshot of your Battle.Net Desktop App so I can see what the issue is.