Saiz was dismissed in stoppage-time for spitting at Newport County's Robbie Willmott as Leeds was knocked out of the FA Cup by the League Two side on Monday (AEDT).
The 26 year-old, cleared of wrongdoing after being accused of spitting by Port Vale manager Michael Brown in August, apologised in a statement issued by Leeds.
Saiz faces a lengthy spell on the sidelines as Leeds, sixth in the Championship, pushes for a return to the Premier League.
"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," Saiz said.
"I would also like to apologise to Newport County 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 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."
Willmott had provided the assist for Newport's winning goal, a corner that was headed home by Shawn McCoulsky, and he accepted Saiz's apology in a social media post.