The 51-year-old was robbed and beaten in the attack near Connaught Place, New Delhi, a police spokesman said.
The woman managed to reach her hotel and the owner called police. A group of men are being questioned about the attack in the district of Paharganj, a popular tourist area near New Delhi railway station which has many backpacker hotels and restaurants.
Danish ambassador Freddy Svane said he was aware of the case but gave no further details.
Police said the woman was walking through the area of narrow lanes and dark streets when she asked two men for directions to her hotel.
Five to six more men then appeared and she was raped, according to senior police official Deepak Mishra.
The tourist was robbed of her cash, iPad and telephone. She refused a medical examination and has since left India to return to Denmark.
Amit Bahl, owner of the Amax Hotel, said: "When she came in, it was terrible." He said the woman was crying and "not in good shape".
Mr Bahl added: "I am really ashamed that this happened."
The rape, which happened on Tuesday afternoon, is the latest case to focus international attention on rape and violence against women in India.
The problem has gained widespread attention since the gang rape and murder of a 23-year-old woman on a bus in December 2012.
Public fury over that case led to more stringent laws that doubled prison terms for rape to 20 years and criminalised voyeurism and stalking.
Four men were sentenced to death for the murder and another was found hanged in his prison cell. A sixth, who was under 18 at the time of the attack, was sentenced to three years detention, the maximum allowed under juvenile law.
But for many women, particularly the poor, daily indignities and abuse continue and the new laws have not made the streets any safer.
Ranjana Kumari, director of India's Centre for Social Research, said India's conservative, patriarchal traditions lead men to use rape as a tool to instil fear in women.
Experts say the rapid growth of India's cities and the yawning gulf between rich and poor are exacerbating the problem of sexual violence, with young men struggling to prove their traditional dominance in a changing world.
Cultural stigmas, police apathy and judicial incompetence have long made it difficult for women to even report rapes but there has been a surge in the number of rapes being reported recently, suggesting women are emboldened to speak up.
Between January and October last year, 1330 rapes were reported in Delhi and its suburbs, compared with 706 for all of 2012, according to government figures.
Last March, a Swiss woman who was cycling with her husband in central India was gang-raped.
These cases threaten India's lucrative tourism industry. Last year, the Tourism Ministry launched an I Respect Women campaign to reassure travellers.
A study by India's Associated Chambers of Commerce and Industry said foreign tourist visits to India dropped 25% in the three months after the rape and murder of the Delhi woman. Tourist visits by women fell by 35%.