Prince Andrew said in a statement he wished to stop "speculation and innuendo" relating to his daughters.
He said it was "complete fabrication" to suggest he wanted titles for any future husbands of the princesses.
He added that continued speculation over a split between himself and Prince Charles was "pointless".
"As a father, my wish for my daughters is for them to be modern, working, young women, who happen to be members of the Royal Family, and I am delighted to see them building their careers," the duke said.
"When they do support the Royal Family in its work this is very much appreciated by my family and, most importantly, by those organisations and to those for whom their participation makes such a difference to their lives."
While acknowledging there is "considerable interest" in the granddaughters of the Queen, Prince Andrew said he "cannot continue to stand by and have the media speculate on their futures based on my purported interventions, which are completely made up".
The statement was issued via the duke's Twitter account.
It came shortly after the princesses' mother, the Duchess of York, urged the media to "stop bullying the York family".
She was speaking as Beatrice, 28, and Eugenie, 26, became patrons of the Teenage Cancer Trust charity.
The duchess and the two princesses visited a specialist teen cancer unit in central London to mark the occasion, meeting young people with the disease.
"Let's focus more on this and less on tittle-tattle gossip," said the duchess, who was divorced from Prince Andrew in 1996.
By Peter Hunt, BBC royal correspondent
Once, personal royal statements were vanishingly rare.
Not anymore.
After Prince Harry leapt to the aid of his girlfriend Meghan Markle, his uncle has used his personal Twitter account to defend his daughters.
And it's significant that the princely views were disseminated this way and not through an official Buckingham Palace statement.
There will be those at the palace who will have questioned the wisdom of this move.
Not least because there has been tension between Prince Andrew and Prince Charles over the future king's plans to slim down the monarchy when his time comes.
Read more from Peter Hunt
The Duke of York's statement was issued in response to reports claiming he intended to persuade the Queen or Prince Charles to make any husbands of Eugenie or Beatrice earls.
In October, there were claims of a row between Prince Andrew and his brother, said to have been sparked by a wish for his daughters to be given more significant roles within the royal family in future.
The statement also comes just over a month after Prince Harry took the decision to issue a statement attacking the media for subjecting his actress girlfriend Meghan Markle to a "wave of abuse and harassment".
There is "no truth" in press claims of a split in the Royal Family over the future roles of Princesses Eugenie and Beatrice, the Duke of York has said.