Yes Call of Duty 3 does, if you meant MW3 then no but it is on ps3, Xbox 360, pc, wii, and even the DS (not 3DS though)
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There are Call of Duty games for the PS2. The last one released was Call of Duty World at War Final Fronts in November 2008. The first 3 Call of Duty games are also available on the PS3 and the highest rated would be Call of Duty 3. See related link for Call of Duty World at War Final Fronts for more information beyond this quote: "the PS2 version is almost completely different with its own set of mission... with no online multiplayer, and no Nazi Zombies game mode, It was developed by Rebellion-instead of Treyarch from the other version-and published by Activision."
The best one would have to be Call of Duty 3 on PlayStation 2. Yes you can go online on it.
No none of the 3 Modern Warfare series games for Call of Duty are made in PS2 versions
Call of Duty Black Op for Nintendo DS Call of Duty Modern Warfare Mobilized for Nintendo DS Call of Duty 3 Roads to Victory Greatest Hits for PSP Call of Duty World at War for Nintendo DS Call of Duty World at War Final Fronts for PS2 Call of Duty 4 Modern Warfare for Nintendo DS Call of Duty 3 for Nintendo Wii, PS3, PS2,Xbox, Xbox 360 Call of Duty Deluxe for PC Call of Duty Finest Hour for PlayStation 2, GameCube, Xbox Call of Duty United Offensive for PC Call of Duty 2 Big Red One for Xbox, Game Cube,PS2 Call of Duty 2 Game of the Year Edition for Xbox 360 Call of Duty 2 Big Red One Enhanced Greatest Hits for PS2 Call of Duty for PC The games that have been rated T are listed above. Since the release of Call of Duty 3 only the portable games have been rated T or Teen and the rest have been rated M or Mature except for the PS2 Final Fronts.
Call of Duty World at War, Modern Warfare, Modern Warfare 2, Black Ops, and Modern Warfare 3 all have online multiplayer.