A similar issue occurred recently through Steam with the digital sale of Spore. That problem seems to have been resolved, and at least one Red Alert 3 player has been able to rectify the current error, though the solution required the game to be re-downloaded directly from EA (complete with SecuROM DRM). Still, EA's eventual intervention in Steam's Spore issue is a possible sign that these Red Alert 3 woes will be alleviated in a more timely manner.
Update: Reader Tom suggests verifying the game cache in Steam (right click in "My Games," and find it under Properties > Local Files), which apparently identifies the files that need to be replaced to enable the game to work properly. Maybe that will help!