Vaughan was left with a badly cut mouth after being caught in the face by Tamas during Albion's 1-0 Premier League win.
The 27-year-old Romanian central defender was retrospectively charged with violent conduct after the FA studied video evidence and match referee Mark Halsey insisted he would have brandished a red card had he seen it during the game.
In a statement on their official website, Albion's legal director/secretary Richard Garlick said: "We have spoken at length with Gabriel and he is adamant that he did not intentionally elbow James in the face.
"However, he admits the way he turned in the penalty area was clumsy and he has apologised to James through a mutual friend for the injury he has suffered.
"We, as a club, also send our best wishes to James. Although we accept Gabriel's version of the incident, both the player and the club have decided against appealing the decision as it would prove very difficult to overturn.”