In my opinion there are no realistic FPS games on the consoles at least nothing like Arma 3 on the PC. I doubt the current consoles could do a game like that justice. Maybe it might be possible on the PS4 or the Xbox 360 successor.
It really depends on what kind of realism a person is looking for. The R6 games (someone mentioned it) were realistic because each mission was supposed to be planned out as it would IRL. The environments were realistic (though graphically dated by today's standard) and the weapons were realistic.
Battlefield is realistic because the environment is always changing, buildings go down providing new paths into enemy strongholds or crushing people inside. Vehicles are realistic, weapons are realistic, environments are based on real environments.
COD looks okay, that is all, nothing about the gameplay itself is realistic other than some of the weapons, and even that is sketchy. The environments are styled after realistic places.
Flashpoint was realistic because the weapons were real, the mission types were real, the tactics are real, and it is based on a tactical simulator that is used by DOD.
That's all I can think of for now, but there are plenty more FPS games, they are a dime a dozen anymore. No one wants an uber realistic shooter, even if they think they do, because they would be filling out reports and creating mission plans, delegating, talking to locals, lol. The simulators we've used actually play more like a FPSRPG because you do quite a but in between firefights, if you do firefights at all. It isn't all run and gun like people who play the videogames think.
