During the closing stages of the 2-1 defeat in South Wales, Saiz received a red card and has acknowledged that his actions in the fixture were "inexcusable".
In a statement, the 26-year-old said: "I would like to take this opportunity to sincerely apologise to the players, staff and supporters of Leeds United for my behaviour in yesterday's game at Newport County AFC.
"I would also like to apologise to Newport County AFC and Robbie Willmott for my actions. I accept the punishment handed to me by The FA.
"My actions are inexcusable, I give my word to all those connected to Leeds United Football Club that I will learn from this situation and never repeat it.
"I am grateful for the support I receive from our fans and I understand that I am a role model to the next generation of Leeds supporters, I will ensure that my behaviour moving forward reflects that."
Earlier this season, Saiz was accused of a similar offence by Port Vale boss Michael Brown, but no evidence was found during the EFL Cup clash in August.
Saiz has netted a total of five goals from 22 appearances in the Championship since signing from Getafe in July.