The Super League club, who suffered relegation following Sunday's 52-26 loss to St Helens, were given the punishment after going into administration.
The Bulls initially appealed the decision in April, and were taken over by a consortium headed by Leeds-based businessman Marc Green, but their on-field troubles continued.
Former player James Lowes replaced Francis Cummins at the helm but was unable to stave off the club's first relegation from the top flight.
"As part of our last press release on the matter, the directors indicated the club would only proceed with legal action in the event it would be of potential benefit to the Bulls," the board of directors said in a statement.
"With a heavy heart and a degree of reluctance the board have therefore taken the decision not to pursue any legal challenge to the points deduction imposed earlier this season.
"For the record, the original appeal against the points deduction was correct and amongst other factors, was as a result of the Rugby Football League’s policy and procedure being flawed."