The man, named only as NT2 in court, wanted a past crime he committed to not come up in search results.
Mr Justice Warby delivered a blow to Google by ruling in his favour on Friday.
The judge rejected a similar claim by a second businessman, referred to only as NT1, who had been convicted of a more serious crime.
The claims were apparently the first of their kind in England.
Carter-Ruck, the law firm which acted for both men, described the cases as “unprecedented” and “groundbreaking”.
Google contested the two claims, which were heard in separate High Court trials in London.
Both businessmen, who cannot be named for legal reasons, were convicted of crimes many years ago.
They argued their convictions were now legally “spent” and they had both been rehabilitated.
Their lawyers said their claims, which were brought under data protection law and for “misuse of private information”, were the first of their kind to be made in England.
Mr Justice Warby said the businessmen complained of Google search engine results that provided links to third-party reports about their convictions.
He said their claims were based on the “right to be forgotten”, or to have personal information “delisted” by the operators of internet search engines.
NT2, who won his case, had been sentenced to six months imprisonment for “conspiracy to carry out surveillance”.
His crime related to complaints about 11 “source publications”.
Mr Warby said an “appropriate delisting order should be made”.
The judge ruled out any damages payment, and told the court: “NT2 has frankly acknowledged his guilt and expressed genuine remorse.
“There is no evidence of any risk of repetition.
“His current business activities are in a field quite different from that in which he was operating at the time.”
NT1 was given permission to appeal after losing his case.
The judge said the information related to his crime “retains sufficient relevance today”.
He had received a four-year sentence for “conspiracy to account falsely”.
Mr Justice Warby told the High Court: “He has not accepted his guilt, has misled the public and this court, and shows no remorse over any of these matters.
“He remains in business, and the information serves the purpose of minimising the risk that he will continue to mislead, as he has in the past.”
The judge explained the men’s names could not be reported because otherwise their claims might be “self-defeating”.
He added it could draw “additional publicity for the information in question”.