Christie, who served as a special assistant to President George W. Bush, said what the White House needs in its communications director is a "seasoned pro" who has a good relationship with the chief of staff.
"Bringing in someone who has not run communications in the White House, I think, is a very foolish move at this juncture," he said.
Plus, Scaramucci will have to deal with a president who likes to constantly tweet, sometimes contradicting the White House's message.
"You can bring in the most seasoned communications professional in the world and have that message blown up by a 140-character tweet," Christie said.
"This president more than ever needs strong people around him, people who are in the position [to say] 'No, Mr. President, it's not good for you to do that. I suggest you take a different course and a different tack."
Scaramucci addressed the issue during his first briefing on Friday. While he wouldn't specifically say what would happen if Trump tweeted something that differed from the White House's message, he said Trump's use of social media has been "very effective" in reaching the American public directly.
"I welcome him continuing to do that. I think it's very, very important to him to express his identity," Scaramucci said.
Meanwhile, it is also important the communications director have a good relationship with the chief of staff, Christie said.
Trump's chief of staff, Reince Priebus, along with top advisor Steve Bannon, resisted the appointment of Scaramucci, NBC News reported on Friday.
As to whether the latest drama will affect Trump's legislative agenda, Christie said tax reform is "in serious peril" unless Trump rolls up his sleeves and gets to work, engaging with members of Congress.
— CNBC's Angelica LaVito contributed to this report.