The protests led to clashes between police and protesters in the heart of the capital, New Delhi, at the weekend.
Singh, who has faced fierce criticism for remaining largely silent about the rape, vowed action against what he called a "monstrous" crime.
The 23-year-old victim, who was attacked and then thrown from a moving bus in New Delhi, remains in hospital in critical condition.
"I appeal to all concerned citizens to maintain peace and calm. I assure you we will make all possible efforts to ensure security and safety of women in this country," Singh said in a televised address to the nation.
Police barricaded roads leading to India Gate, an imposing Arc de Triomphe-style war memorial in the centre of the city, that has become a hub of the protests by mostly college students. Many metro rail stations in the fog-shrouded city were also closed, hampering movements around the city of 16-million.
The uproar over the rape and what many protesters and political commentators see as a slow and ineffectual response by the government overshadowed an official visit by Russia's President Vladimir Putin.
Indian officials said a joint news conference by Singh and Putin at a venue close to India Gate would be moved and held at the prime minister's residence instead.
The spasm of violence at the weekend saw police use batons, teargas and water cannon against protesters around the city. Protests have also taken place in other Indian cities but they have been more peaceful.
The ferocity of the protests in the capital, many driven by social media, appear to have caught the government by surprise. Singh made his first comment on the rape in a statement on Sunday, a week after the event.
New Delhi has the highest number of sex crimes among India's major cities, with a rape reported on average every 18 hours, according to police figures. – Reuters