None of the Call of Duty series of Games are Move Compatible so you can not do it. Only Move games and Move compatible games can use the PS3 Move Controller.
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Modern Warfare 2, the online play is beast.
Yes Call of Duty 4 Modern Warfare is still out for the PlayStation 3
computer, ps3 or xbox360
Yes you can you would need to send them a game invite by press the middle button on your controller and going to your friends list. You will need to be in your own game lobby before doing this
Modern Warfare 2 will be released on November 10, 2009. It is the sequel to Call of Duty 4: Modern Warfare and it is the 6th game in the Call of Duty series.