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The winger, 22, was dismissed in the loss to St Johnstone after appearing to react to Stephen Mitchell's award of a foul throw.
Tynecastle head coach Ian Cathro said after the game he would review the incident, but backed his player.
But BBC Scotland has learned the club is not planning to lodge an appeal.
Nicholson will be suspended for Saturday's home match against Dundee.
"I want to see the footage [of Nicholson's red card]," Cathro said in his post-match interview following Wednesday's defeat in Perth.
"I don't think there was any aggression directed in that. I take that quite clearly from how Sam describes the situation and from other players around it. That's something that needs looked at."
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Hearts will not appeal against the red card Sam Nicholson received for an alleged spit in the direction of an assistant referee at McDiarmid Park.