Meghan, 39, sued Associated Newspapers over the publication of five articles that reproduced extracts of a handwritten letter sent to her estranged father, Thomas Markle, 76.
She was last month granted a summary judgment, a legal step that saw the privacy claim and the bulk of the copyright claim resolved in her favour without a trial, prompting her to claim a victory over "moral exploitation".
At a remote hearing on Tuesday, Lord Justice Warby heard further arguments on "the next steps" in the legal action, noting wryly: "Everything seems to be in dispute."
The hearing was scheduled to deal with applications for Associated to pay the duchess' legal costs as well as the remaining parts of her claim, including unresolved issues relating to copyright and her request for an injunction to prevent further publication of the letter.
Associated listed 10 grounds of appeal, raising the prospect that the case may yet go to trial.
Among them, the newspaper group argued that the judge had failed to heed a precedent set in a 2013 privacy case involving the Daily Mail, Johnson's former lover, Helen Macintyre, and their then 3-year-old daughter.
In that case, the Court of Appeal ruled that the public had a right to know Johnson, who at the time was Mayor of London, had a "brief adulterous affair" with a woman who later gave birth to their daughter.
Macintyre, who lost her battle to keep the paternity of her daughter secret, was a professional art consultant named only as "AAA" in public court documents.
Associated said Lord Justice Warby's approach was "inconsistent" with the appeal court's decision in that case, as allegedly private information had been disclosed before and after the relevant newspaper articles and were deemed to "reduce or weaken or compromise her reasonable expectation of privacy".
It said in written submissions: "The judge in the present case made no such findings, and his conclusion a trial was unnecessary meant he was unable to do so.
"The exercise was the more important in this case since the account allegedly given by the claimant's friends on her behalf and caused by her was given to the public and pleaded as 'one-sided and/or misleading' in several significant and specified respects."
Associated also argued that Lord Justice Warby gave "excessive emphasis" to the duchess' privacy rights at the expense of its rights of freedom of expression and failed to give due weight to her attitude towards her own privacy, including the information released via her five friends to People magazine and to the authors of Finding Freedom.
It was an article headlined "The Truth About Meghan" published in People magazine and based on interviews with five of Meghan's friends, that made the first public reference to the letter.
At the remote hearing, Ian Mill QC, for the duchess, asked the High Court to order Associated to hand over any copies of her letter to Thomas Markle and to destroy any electronic copies or any notes made about it.
Mill also applied for an injunction to "restrain the acts of copyright infringement and misuse of private information" and sought an order requiring Associated to publish a statement about the duchess' legal victory on the front page of the Mail On Sunday and the home page of MailOnline.
He indicated that the duchess was willing to "cap her damages" for misuse of private information "at a nominal award", in order to "avoid the need for time and cost to be incurred in debating these issues".
After hearing submissions from both sides, Lord Justice Warby ordered the newspaper to make an interim costs payment of $860,000 and said there should be "an account of profits" in relation to damages for the breach of copyright.
He granted the duchess, who is expecting her second child, "a final injunction restraining misuse of private information".
Lord Justice Warby declined to order the "delivery up or destruction" of any copies of the duchess' letter "at this stage". He will hand down a written ruling on the request for a front-page statement on Meghan's victory but warned it would be "considerably more limited than the order sought".
Any "financial remedies" for misuse of private information will be considered at a future hearing, which will also address the duchess's data protection claim and "the issue of copyright ownership".
The judge dismissed the newspaper's application to appeal but said: "The Court of Appeal, of course, might take a different view."
An Associated spokesperson said last night that it was "disappointed" with the judge's decision, adding: "We will be applying to the Court of Appeal for permission to appeal in relation to his decisions on both the privacy and copyright claims."
In court documents lodged on her behalf, the duchess criticised the newspaper's refusal to accept that it had lost both privacy and copyright claims, as well as its insistence that the claim for breach of data protection should go to trial.
She indicated that she was not seeking to pursue her outstanding claim for breach of her data protection rights, the copyright claim and the claim for infringement, stating that success on these points "would not add materially to the relief to which she is already entitled".
Her lawyers said she wanted the remainder of the case to be dealt with "in as sensible and proportionate a manner as possible".
They said that as Associated Newspapers appeared intent on forcing her to continue her data protection claim, she would have "no alternative" but to apply for summary judgment.
Antony White QC, for Associated, said the request for a front-page statement was unnecessary in light of the "very extensive coverage" it afforded, adding that the application seemed to be "intended more as a species of punishment or retribution, rather than as a necessary and proportionate measure in the interests of the claimant or the public".
The barrister said the duchess' "extremely large costs bill" was disproportionate.
