The order was passed by the division bench of Justices S C Dharmadhikari and Gautam Patel on Thursday after reading a report filed by the dean of the Government-run J J Hospital in the case.
Protection of Rights Association, an NGO, has filed a petition alleging that medical negligence was responsible for the boy's death.
The court asked the Samata Nagar Police Station to complete the probe and file final report within 30 days.
"His parents have been running from pillar to post but neither police nor forensic lab at suburban Kalina have responded," said advocate Aniket Nikam, the NGO's lawyer.
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"It is only after the court's intervention that police filed an FIR last month," he said.
The petition seeks that the forensic lab should reveal the cause of death.
The boy had been operated for heart condition at a Delhi hospital when he was 7-month old. In 2011, he developed a breathing problem, and a further surgery was advised.