Prince Nikolai, 23, Prince Felix, 20, Prince Henrik, 13, and Princess Athena, 10, will lose their titles from January and will instead be referred to as Count and Countess of Monpezat, the palace announced in September.
The official reason for removing their titles was to allow the children of the youngest son of the queen, Prince Joachim, to lead more normal lives.
But Joachim said he had been caught unaware by the “very sad” announcement that his daughter and three sons would no longer be princes and princesses.
The queen, 82, has now issued a statement on her website apologising for the “strong reaction” to her decision.
She wrote: “I have made my decision as queen, mother and grandmother, but, as a mother and grandmother, I have underestimated the extent to which my younger son and his family feel affected.
“That makes a big impression, and for that I am sorry. No one should be in doubt that my children, daughters-in-law and grandchildren are my great joy and pride.
“I now hope that we as a family can find the peace to find our way through this situation.”
Prince Joachim, 53, has four children from two marriages: Nikolai, Felix, Henrik and Athena, ranging in age from 23 to 10.
Last month the palace said that the queen “wants to create a framework for the four grandchildren, to a much greater degree, to be able to shape their own existence without being limited by the special considerations and obligations that a formal affiliation with the royal house as an institution implies”.
It added: “The queen’s decision is in line with similar changes that other royal houses have carried out in recent years in different ways.”
The mother of Prince Joachim’s two eldest sons told Danish media she was “shocked” by the decision. “This came from out of the blue. The children feel excluded,” Countess Alexandra told the daily BT. “They can’t understand why their identity is being taken from them.”
The queen’s four other grandchildren, born to Crown Prince Frederik, 54, will retain their titles but when they come of age only the future king, Prince Christian, will receive a grant, a decision taken in 2016.