Manager Mick McCarthy reiterated his belief that Henry should not have been dismissed by referee Andre Marriner for his tackle from behind on Gunners midfielder Tomas Rosicky.
But he added that the advice he has received over the weekend suggests any appeal is unlikely to succeed, and that Henry's three-match ban could be extended.
"Having spoken to a lot of different people over the weekend, including a number of ex-referees, I haven't found anyone yet who thinks that Karl should have been sent off," McCarthy told the club's official website.
"But having also sought advice in terms of the appeal process, we have decided to accept the punishment as it stands."
Henry will therefore miss Sunday's home game with his former club Stoke as well as subsequent clashes with Fulham and Blackburn.
McCarthy's key concern is that the Football Association's regulatory commission require evidence of a "serious and obvious error" before upholding an appeal for wrongful dismissal.
He continued: "Under that wording, I have been advised that any appeal would be very unlikely to succeed.
"And the commission also have the right to lengthen any suspension to twice the initial ruling if they believe an appeal was made which didn't have any chance of success.
"Despite our frustration and disappointment at losing Karl for three games, we can't run the risk of that ban being extended and so unfortunately we have to take our medicine and move on."