The rapist, who cannot be named as he was a minor at the time of the crime, was sentenced to a maximum three years in a reform facility in August 2013.
He was released from detention recently and is currently housed with a charity because of fears over his safety.
Under the current law, his detention cannot be extended, the court said.
Jyoti Singh, a 23-year-old physiotherapy student, died after being brutally raped by six men on a moving bus in 2012, causing global outrage.
Four adult convicts in the case are appealing against death sentences. A fifth died in prison.
The release of the youngest convict has been opposed by many people, including the parents of Ms Singh.
In a last-ditch attempt to prevent his release, the Delhi Commission for Women chief Swati Maliwal filed a petition in the Supreme Court late on Saturday night.
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On Monday, the Supreme Court dismissed the petition, saying it "shared" the concern of most citizens but its hands were "tied" by the law.
"We share your concern but our hands are tied by the existing law. There has to be clear legislative sanction to extend the detention period beyond three years. Under the present law, detention cannot be extended beyond three years," the top court said.
The BBC's Geeta Pandey in Delhi says the Supreme Court order was a foregone conclusion - the convict had already served the maximum three year sentence under India's current laws dealing with juvenile crime and he cannot be kept in detention any longer.
Last week, a legal challenge by politician Subramanian Swamy to stop the release had also failed.
The Delhi high court ruled on the case on Friday, saying: "We agree it is a serious issue. But after 20 December, the juvenile cannot be kept at a special home per law."
Although the convict is now an adult, he has been handed over to a charity, where he will remain for at least two years.
16 December 2012: A 23-year-old physiotherapy student is gang-raped by six men on a bus in Delhi, her male friend is beaten up and the pair are thrown out after the brutal assault
17 December: Key accused Ram Singh, the bus driver, is arrested. Over the next few days, his brother Mukesh Singh, gym instructor Vinay Sharma, fruit seller Pawan Gupta, a helper on the bus Akshay Thakur, and the 17-year-old juvenile, who cannot be named, are arrested.
29 December: The victim dies in hospital in Singapore from injuries sustained during the assault; body flown back to Delhi
30 December: Victim is cremated in Delhi under tight police security
11 March 2013: Ram Singh dies in Tihar jail; police say he hanged himself, but defence lawyers and his family allege he was murdered
31 August: The juvenile is found guilty and sentenced to three years in a reform facility
13 September: The four adult defendants are convicted and given the death penalty by the trial court
March - June: The convicts' appeal in the Supreme Court and the death sentences are put on hold
December 2015: Youngest convict released from correctional centre after serving maximum sentence allowed
India's Supreme Court has dismissed an appeal against the release of the youngest convict in the notorious 2012 Delhi gang rape case.