Are you using the free weekend of MW2 on Steam or the fully bought version that won't expire?
If it still doesn't work after reinstalling, etc. try this:
1) Go to "My Games" & right click on the game (either being Modern Warfare 2 or Modern Warfare 2 Multiplayer) & click on "Properties."
2) Go to the "Local Files" tab & click on "Verify Integrity of Game Cache..." & then let it finish testing to see if all your files are there. If a file is missing, it will automatically re-download that missing file for you after it has finished testing.