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Rooney was given a second yellow card for throwing the ball in anger towards referee Phil Dowd during the 2-0 defeat at Craven Cottage.
Scholes was sent off early on for handling the ball on the line.
Both men will miss United's next match, at home to Aston Villa, on April 5.
The FA is to write to Rooney to issue him with a formal warning after he punched the corner flag before leaving the pitch, but the England striker was not reported for his angry words towards Dowd following his send off.
"We have clarified with Phil Dowd that there was no abuse from the player after he was dismissed," an FA spokesperson was quoted as saying.
"In relation to the incident when he punched the corner flag, the referee did not see that and we have written to Wayne Rooney reminding him of his responsibilities."
United defender Nemanja Vidic is also suspended for the Villa game as he serves the second match of a two-game ban for his sending-off in the 4-1 defeat by Liverpool 10 days ago.
