Lord McAlpine's lawyer said Sally Bercow "must now accept" that her tweet about the Conservative peer was defamatory after the High Court ruled it was libellous.
Andrew Reid of RMPI Solicitors said:
The apologies previously received from Mrs Bercow did not concede that her tweet was defamatory. Clearly she must now accept this fact.
The failure of Mrs Bercow to admit that her tweet was defamatory caused considerable unnecessary pain and suffering to Lord McAlpine and his family over the past six months.
Mr Justice Tugendhat's judgement is one of great public interest and provides both a warning to, and guidance for, people who use social media.
The High Court judge who ruled a tweet by Sally Bercow about Lord McAlpine was libellous said her followers - who numbered 56,000 - would likely know by the time of her post the elements of the story told on Newsnight.
"In my judgement, the reasonable reader would understand the words 'innocent face' as being insincere and ironical", Mr Justice Tugendhat said.
"There is no sensible reason for including those words in the tweet if they are to be taken as meaning that the defendant simply wants to know the answer to a factual question".
The reader would reasonably infer that Mrs Bercow had provided "the last piece in the jigsaw".
Her tweet, by implication, was a repetition of the accusation with the addition of the name which had previously been omitted.
"It is an allegation of guilt. I see no room on these facts for any less serious meaning", the judge said.
Sally Bercow said the High Court's ruling that her tweet about Lord McAlpine was libellous "should be seen as a warning to all social media users".
Mrs Bercow said in a statement that she did not write the tweet "with malice" and "did not intend to libel" the Conservative peer.
– A statement by Sally Bercow
I was being conversational and mischievous, as was so often my style on Twitter.
I very much regret my tweet, and I promptly apologised publicly and privately to Lord McAlpine for the distress I caused him. I also made two offers of compensation.
Lord McAlpine issued proceedings and the last few months have been a nightmare. I am sure he has found it as stressful as I have. Litigation is not a pleasant experience for anyone.
Sally Bercow said she has "accepted an earlier offer" made by Lord McAlpine's lawyers "to settle this matter" after the High Court ruled a tweet she wrote about the Conservative peer was libellous.
Sally Bercow said she was "surprised and disappointed" after the High Court ruled a tweet she wrote about Lord McAlpine was libellous.
Mrs Bercow added, "However, I will accept the ruling as the end of the matter. I remain sorry for the distress I have caused Lord McAlpine and I repeat my apologies".
A tweet by Commons Speaker's wife Sally Bercow about Lord McAlpine was libellous the High Court in London ruled today.
Mr Justice Tugendhat will give his decision on the meaning of an allegedly libellous tweet she sent after a Newsnight report that wrongly implicated the Tory peer in child sex abuse allegations.
Earlier this month the judge reserved his decision in the case.
If the judgment goes in Lord McAlpine's favour, there will be another hearing at a later date on the appropriate level of damages, unless the two sides reach a settlement.
Mr Justice Tugendhat has reserved his decision to a later unspecified date in the Sally Bercow tweet case.
A tweet by Commons Speaker's wife Sally Bercow pointed "the finger of blame" at Tory peer Lord McAlpine during a media frenzy over allegations of child sex abuse, the High Court has been told.
Ms Bercow has always denied that the tweet was defamatory.
Sir Edward Garnier QC said it would be difficult to think of a more serious meaning than the one advanced by Lord McAlpine, who was not in court.
He told Mr Justice Tugendhat that his case was founded on the circumstances in which Mrs Bercow decided to tweet the peer's name - the media frenzy over a story that spread "like wildfire".
"The tweet, by itself, suggests that 'Lord McAlpine' has done something wrong. Drawing attention to someone and then adding the expression 'innocent face' hints at wrongdoing and negates any suggestion that the tweet was a neutral query to which the defendant was looking for an answer.
"The inclusion of the words 'innocent face' was giving a nudge and a wink to readers."
Mrs Bercow's counsel, William McCormick QC, told the judge that she promptly tweeted her apologies, provided letters apologising for the distress caused and making clear that the underlying allegations were untrue, and has offered to settle the case.