Stevenson was shown a straight red card by referee Willie Collum for a first-half foul on Gary Mackay-Steven, catching the winger with a sliding studs-first tackle as he stretched to reach a loose ball.
After the match, Hearts manager John McGlynn said: "According to Ryan, the lad says that he never touched him but that is the rulebook and he has to go with the rulebook.
"The players feel if you go into a tackle when you are on the ground you can hardly go with your studs on the ground."
However, in a brief statement, the club said: "Hearts can today reveal the club will not be appealing Ryan Stevenson's red card against Dundee United."
Stevenson will now serve a three-match ban, missing Clydesdale Bank Premier League games against Kilmarnock, Inverness and St Mirren.
The former Ayr midfielder was last month handed a retrospective two-match ban by a Scottish Football Association judicial panel for a challenge on Hibernian captain James McPake, which Collum did not punish during the game.
- Sports & Recreation
- Ryan Stevenson
- Willie Collum