Grondona, who is also president of his country's national association, branded the English "liars" and "pirates" at the FIFA Congress in Zurich in June, but has now written a letter of apology.
Bernstein told the Leaders in Football conference in London that the FA's relationships with FIFA and UEFA were improving, and that Grondona had retracted his remarks.
He said: "I was pleased to receive a letter of apology from the Argentinian (FA) president Grondona following his unacceptable comments about England in Zurich."
Grondona's comments came after Bernstein had asked the FIFA Congress to postpone the FIFA presidential election where Sepp Blatter was elected unopposed.
The Argentinian told the Congress: "We always have attacks from England which are mostly lies with the support of journalism which is more busy lying than telling the truth. This upsets and disturbs the FIFA family.
"I see it at every Congress. They have specific privileges with four countries having one vice-president.
"It looks like England is always complaining so please I say will you leave the FIFA family alone, and when you speak, speak with truth."
In an interview with a German press agency, Grondona had earlier called England "pirates" and added: "With the English [2018 World Cup] bid I said: 'Let us be brief. If you give back the Falkland Islands, which belong to us, you will get my vote'. They then became sad and left."
Bernstein said there was "great disappointment" over the pace of FIFA reforms in the wake of the corruption scandals. He also insisted that he had no regrets about standing up at the FIFA Congress to call for the election to be postponed.
He added: "We had a UEFA strategic meeting two weeks ago in Cyprus and there was great disappointment expressed there about the lack of progress that FIFA are making in terms of governance.
"There is a FIFA ExCo (executive committee) this month and we are hoping something will come out of that but I wouldn't hold your breath.
"Running 210 nations or so is not an easy job. UEFA has influence but is one fifth of FIFA and every nation has one vote. It's a difficult nut to crack, we will just have to see what Mr Blatter will do."
Asked about his intervention at the FIFA Congress, Bernstein admitted that UEFA had tried to stop him doing so.
He added: "I wouldn't say it was foolhardy, there was a momentum of events and I felt I had to say something.
"It was also a question of free speech. The morning of that session UEFA held an emergency meeting to discourage me from saying something.
"I said that within English football there are plenty of disagreements but everyone has the right of free speech. I would definitely do it again."
Bernstein also promised the FA would get its own house in order after the recent Parliamentary Select Committee inquiry into Football Governance urged the Government to legislate if the game's governing bodies failed to adopt new practises.
Bernstein said: "If we can't then there is a danger that they will legislate.
"Our job - and that's why we're working so closely with all our friends at the Premier League and Football League and the national game and others - is to make sure that doesn't happen."
He added: "The Select Committee report is not perfect. I think there are parts of it that leave something to be desired.
"But there is merit within there.
"We have to pick the best bits, work out a package that we can deliver together and come up with something so that the Government doesn't have to get involved."
One Select Committee criticism centred of the lack of diversity within the FA, who are set to appoint two new independent directors to their board.
Quizzed about the prospect of a woman being selected, Bernstein said: "A woman on the board would be wonderful if we could find the right woman.
"In all these things, we want the best people.
"I don't think we should be absolutely governed by definitions.
"If the best includes diversity, I think that's fantastic."
Bernstein did admit more ex-players were needed within the organisation, saying: "We should see more footballers coming through the organisation.
"I think it would be fantastic."