The tennis player organized a damage-control conference when the scandal hit the headlines.
「謝罪会見」は、damage-control press conference (英訳3)
「記者会見での謝罪」は、press conference apology (英訳2)
「謝罪会見をする」は、give a public apology at a press conference (英訳1)
と表現できます。
hold a press conference to apologize to the public
They apologized to the public in front of the media
They publicly apologized to the office(victim) in front of the media
一言で言い換える言葉はたぶんないので
hold a press conference to apologize to the public （公式に謝罪するための記者会見を開く）
They apologized to public in front of the media（メディアの前でみなさんにお詫びした）
They publicly apologized to the office(victim) in front of the media （メディアの前で事務所に（被害者に）公に謝罪した。
英語にするときは、「誰が」「誰に」「何をしたのか」をはっきりさせておくのが重要です。不祥事を起こした人が記者会見を開くのは公式に謝るためということで、書いたのが最初の文。世間をお騒がせしましたと世間に対して謝るのだからと書いたのが二つ目。
三つ目は謝る先は皆様ではないけど、謝る様子を公に報道させると言う形だったと言いたいときを想定したのが三つ目。
この他に、She admitted her mistake (fault)in front of the camera「カメラの前で自分の誤りを(過ちを）認めた」というような言い方も出来ます。
he apologized publicly for the treatment of refugees at a press conference.
The first example displays a summary of the apology made to the public and in front of the press so that it is communicated in the media beyond the people that are present.
The second answer gives an example of how it could be used in a sentence. a man shows sorrow and regret for the treatment endured by a group of people.
The press conference has made their apology public live on TV today.
A public apology was made by the press conference.
->*The press conference has made their apology public live on TV today.
->*A public apology was made by the press conference.
--->> Both answers are good to use and it clearly indicates that the apology was made in public.
The press conference has made their apology public live on TV today.
（本日記者会見で、謝罪をテレビで公開しています。）
A public apology was made by the press conference.
（公式な謝罪は記者会見で行われました。）
両方の回答のは良い回答の仕方です。
また、その謝罪が公開されたことを明らかに示しています。
examples
"The statesman had to address the press and give a formal apology."
"I would like to give a formal public apology."
or
"I would like to apologise publicly."
"The statesman had to address the press and give a formal apology."
その政治家は、マスコミに対処して、謝罪会見をしなければならなかった。
"I would like to give a formal public apology."
謝罪会見をしたい。
"I would like to apologise publicly."
謝罪会見したい。
The footballer issued a public apology for his behaviour off the pitch.
The government have issued a public apology for the mess up caused by Brexit.
The footballer issued a public apology for his behaviour off the pitch.
（そのサッカー選手はピッチ外での自らの行いについて公に謝罪しました）
The government have issued a public apology for the mess up caused by Brexit.
（政府はブレグジットによる混乱について公に謝罪しました）
アメリカ合衆国 
イギリス 
日本 
南アフリカ 
