Unfortunately, Modern Warfare 2 is powered by the Ricochet anti-cheat program, which doesn't work on operating systems beyond Windows. Users will not be able to play Modern Warfare 2 on Steam Deck, which runs on Linux-based SteamOS. However, one can embrace a few workarounds to counter this limitation.
I've been playing Call of Duty Modern Warfare 3, along with Modern Warfare 2 and Warzone, on my Steam Deck. And guess what? This isn't running on Windows; it's all happening on Steam OS.