The man is paying the woman around £200,000 a year for the boy's upkeep.
But the woman had asked a judge to increase the figure to nearly £800,000 a year.
She had told Mr Justice Bodey she wanted a "vast income provision" so that the youngster could "replicate" his father's lifestyle.
The judge said the woman wanted her son to have boxes at the Emirates Stadium - where Arsenal play - and at Ascot racecourse. She also wanted him to have membership of Ascot and Wentworth golf clubs as well as money for domestic staff and first class flights.
But Mr Justice Bodey has ruled against her. He said £200,000 a year was enough. The judge had analysed the case at a hearing in the Family Division of the High Court in London.
He said the man was in his 50s and the woman, who grew up in an ‘affluent’ Middle Eastern family, in her 30s. And he said the boy lived with his mother in London.
‘The parties never lived together,’ said Mr Justice Bodey in his ruling. ‘The father has never seen (the boy), nor expressed any interest in him. The mother, therefore, bears the entire responsibility of caring for him and bringing him up. She wishes to do so in a lifestyle commensurate with that of the son of a member of a hugely wealthy Royal Family.'
Mr Justice Bodey said the man appeared to rank ‘pretty clearly among the super-rich’ and seemed to move ‘naturally within a world of opulence’.
‘Unhappily the mother has a fervent belief that (the boy) and therefore she herself, by way of a carer's allowance, should be allowed a vast income provision to enable (the boy's) lifestyle to replicate that of the father; but almost without regard to the fact that he is a seven-year-old growing up in London,’ said the judge.
‘(She) seeks a box at the Emirates and at Ascot for (the boy), together with membership of Ascot and Wentworth golf club ... (She) refers to needing a nanny/governess/house manager and a housekeeper. She seeks air travel for herself and (the boy), first class return twice a year ... and for economy flights for a nanny/governess.’
Mr Justice Bodey said he was satisfied the current amount the man was paying was ‘suitable and appropriate’.
He said he had decided that it would be ‘just’ to dismiss the mother's application.
The judge said the man and woman had been embroiled in family court litigation for several years.
He said lawyers' bills ran to ‘several million pounds’ - almost all of which had been paid by the man.