Kompany was dismissed for a professional foul on Nikica Jelavic in the 10th minute of City's 2-0 win at the KC Stadium in the Barclays Premier League.
The Belgian reacted angrily to the decision by referee Lee Mason and kicked a wall as he made his way down the tunnel. He also made a gesture towards fourth official Anthony Taylor that could be interpreted as offensive.
It is understood that the FA has since contacted Kompany for his observations and decided, after also studying Mason's report, that no further action will be taken.
That means Kompany will now serve his mandatory one-match suspension for the red card in this weekend's home clash against Fulham and then be eligible for the derby against Manchester United on March 25.
City goalkeeper Joe Hart is also facing no further action over his role in the game's second major flashpoint.
Hart was involved in an angry confrontation with Hull's George Boyd in which the pair clashed foreheads.
As England international Hart was booked by Mason at the time, the FA considers that matter - in line with its own rules - to have been dealt with.
Boyd, however, is facing a misconduct charge after pictures appeared to him spitting in the direction of Hart during the exchange.
The clash occurred after Boyd was denied a penalty by Mason following what he felt was a foul by Hart.