The public apology came after comments deemed "discriminatory on grounds of race" were made to them by former England women's coach Mark Sampson.
"I hope that the FA learns lessons from this whole sorry saga to ensure that all in the sport and the wider public have faith in their processes and procedures," she said.
"We have been clear that we expect world leading standards of governance from all our national sports governing bodies."
Senior FA officials came under fire for the affair during a parliamentary inquiry in which Aluko said the body had had an "agenda" to protect Sampson and its reputation.
An investigation by an independent barrister, which prompted the FA apology, also concluded that the 35-year-old Sampson, who was sacked by the FA last month for "inappropriate and unacceptable" behaviour in a previous job, was not racist.
Chelsea's Aluko also made the claim that the FA's chief executive Martin Glenn had intimated to her that if she released a statement saying the FA was not institutionally racist then she would receive the second part of an 80,000-pound settlement she had agreed with them.
"I felt that was bordering on blackmail. I categorically refused to write any statement," she said.
Glenn categorically denied the accusation.
The Football Association is one of a number of sporting bodies in England to come under scrutiny over its governance.
"This is why, alongside Sport England and UK Sport, we have drawn up a new Code for Sports Governance that publicly funded sports bodies, including the FA, must adhere to," Crouch said.
(Reporting by Martyn Herman, editing by Pritha Sarkar)
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