A 68-year-old woman from Delhi has been confirmed as the second Indian to die from the coronavirus.
The woman, who had underlying health conditions, is thought to have been infected by her son who travelled to Switzerland and Italy last month.
India's first fatality from the virus was confirmed on Thursday.
The 76-year-old man, from the southern state of Karnataka, died after returning from a month-long visit to Saudi Arabia on 29 February.
People who came in contact with the man are being traced and quarantined, the state's health minister said. India has 82 confirmed cases of the virus, the health ministry says.
The Delhi woman's son was "initially asymptomatic but developed a fever and cough after one day", a government statement said. The family were then screened and the mother and son admitted to hospital.
The 76-year-old Karnataka man was screened at the airport on his return but showed no symptoms at the time. After he developed difficulties last week, he was taken to hospital. He died on Tuesday but it was not reported until Thursday.
India's Supreme Court has said it will only hear urgent cases from Monday, and has restricted the number of people who can enter a courtroom.
Karnataka has banned all gatherings including weddings, sports events and conferences for a week as the country attempts to slow the spread of the virus.
Malls, movie theatres, pubs and night clubs have also been shut.
"The government will decide on further action after a week following a review," the state's chief minister BS Yediyurappa announced on Friday.
But he said that government offices would continue to function as normal.
India has taken a number of steps to halt the spread of Covid-19:
- All visas, barring a select few categories, have been suspended for a month
- Visa-free travel afforded to overseas citizens of the country has been suspended until 15 April and even those allowed in could be subject to 14 days of quarantine
- Schools, colleges and movie theatres in the capital, Delhi, have been shut until 31 March
- The Indian Premier League (IPL), featuring nearly 60 foreign players and scheduled to begin on 29 March, has been postponed to 15 April
- Two one-day cricket matches between India and South Africa will be played behind closed doors
India's health ministry says it was among the first countries in the world to prepare for an outbreak of the respiratory illness, and denied allegations that it was slow in testing suspected cases.
"Our surveillance system is strong and we are able to quickly identify any symptomatic patients," RR Gangakhedkar from the Indian Council of Medical Research (ICMR) told reporters on Thursday.
However, there are concerns about whether the country will be fully equipped to prevent and treat an outbreak.
It would be near impossible for India to force its citizens into mass quarantine and hospitalise people in numbers like China, says the BBC's Soutik Biswas.
Our correspondent says there are also concerns about the country's poor healthcare data. India has a shoddy record in even recording deaths and disease - only 77% of deaths are registered, and doctors are more likely to get the cause of death wrong than right, according to a study the Toronto-based Centre for Global Research. There is patchy data for flu-related deaths.
Rumours, myths and misconceptions spread through social media could also hamper an effective response to the infection.