Bees striker Elder was stretchered off following an aerial challenge with Mills during Saturday's goalless draw at the Don Valley Stadium and is still in hospital with a fractured eye socket and cheekbone. Match referee Karl Evans took no action at the time of the incident, but Brentford later wanted the former Derby player punished.
An FA Spokesman said: "We can confirm that we have written to Rotherham and informed them that we are unable to take any disciplinary action against the Rotherham United player Pablo Mills.
"In accordance with Law 5, FIFA prohibits further disciplinary action from being taken in respect of an incident where it was seen and dealt with at the time by the referee.
"The match referee Mr Karl Evans has reviewed the incident on video and remains of the view that the conduct of the Rotherham United player did not merit a dismissal and the evidence available does not provide sufficient grounds to conclude that the player should have been dismissed."
Rotherham's chief operating officer Paul Douglas said: "We are pleased with the ruling, but everyone at the club would like to reiterate that our thoughts and best wishes are with Nathan and we hope he makes a speedy and full recovery from his injury."