United appealed last week after McNamara was given an immediate three-game ban - with two more suspended - after a touchline spat with Tommy Wright, the St Johnstone manager.
That allowed McNamara to take his place in United's semi-final victory over Rangers but, with five games left in the Scottish Premiership season, the appeal risked delaying the former Celtic player's ban until the May 17 final with the Perth club.
A club statement read: "We firmly believe that there are solid grounds to appeal the sentence imposed. However, having given the matter further thought and as no hearing date has yet been fixed, we now believe it is in the best interests of all concerned to accept the sentence imposed and withdraw the appeal."
McNamara was found guilty of misconduct by the "repeated use of offensive, insulting and abusive language" while Wright was given a suspended one-match ban for "leaving the technical area in the absence of special circumstances and by adopting a threatening and aggressive attitude towards an opposing member of team staff".
Wright said the altercation had not soured his relationship with McNamara. "Jackie and I had a drink afterwards and we met up and had a coffee at the Hibs match recently," he said. "Life's too short told hold grudges and Jackie has said the same. We've spoken a lot since I came here and there's a lot of respect between us."
St Johnstone and United meet on league business tomorrow and neither manager believes the game will have an influence on the cup final. "This weekend won't give anyone a psychological edge - our semi-final game against Aberdeen showed that," Wright said.
"What happens on Saturday won't have any bearing on it. Obviously people will be thinking about the final, it's only human nature. But, as a coaching staff, if we feel anyone is thinking about it too much then we have to put a stop to it."
McNamara said: "I don't think we should be reading anything in this game as regards the cup final. It is a different stadium and pitch and it will be different from what the final will be like. There is a feelgood factor because we have a final to look forward to.
McNamara was thrilled to see three of his players, Stuart Armstrong, Ryan Gauld, and Andrew Robertson, nominated for the Scottish PFA Young Player of the Year Award.
"I think it is great for us to have the youngsters mentioned like this. I also want to show that it was the right choice to give me the job. Being involved in a cup final and hopefully going on to win it allows me to try to do that. That can repay the club for the confidence they showed in me by bringing me here."