Cordeiro apologised on Wednesday for language in a court filing made by the federation in a legal dispute with its women's national team over pay equity.
The filing said men's national team players had a greater level of responsibility than the women and that their job "requires a higher level of skill based on speed and strength".
On Wednesday, members of the U.S. women's' team took the field for their third and final SheBelieves Cup game against Japan with their warm-up jerseys worn inside out to obscure the U.S. Soccer logo in protest.
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After the match, which the United States won 3-1, co-captain Rapinoe said the language in the court filing reflected "blatant misogyny and sexism."
"That [statement] wasn't for us," she said post-match. "That was for fans, for the media, for sponsors, because that all sounded pretty similar to what we've heard before.
"You want to talk about hostility? Every negotiation we've had, those undertones are in there, that we're lesser - every mediation that we had, every time we meet with them and obviously the reason that we filed this lawsuit.
So for him to put that out saying sorry presumably to us, we don't buy it.
She added: "Is that truly how they feel about 50% of the population that they are supposed to be stewarding into the game of soccer?
"I don't think anyone wants to sponsor an organisation that is being blatantly misogynistic and sexist.
"I know that we're in a contentious fight but that crossed a line completely."
Sponsor Coca-Cola told the Wall Street Journal it found the language used in the filing "unacceptable and offensive."
"I sincerely apologise for the offence and pain caused by language in this week's court filing, which did not reflect the values of our federation," Cordeiro said in a written statement.
"Even as we continue to defend the Federation in court, we are making immediate changes.
"I have made it clear to our legal team that even as we debate facts and figures in the course of this case, we must do so with the utmost respect."
The women's team sued the U.S. Soccer Federation for gender discrimination just over a year ago in a lawsuit that included complaints about wages and working conditions.
A trial date has been set for May 5 after talks between the two parties broke down.
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