Deaths grew by a record 4,205 while infections rose 348,421 in the 24 hours to Wednesday, taking the tally past 23 million, health ministry data showed. Experts believe the actual numbers could be five to 10 times higher.
Funeral pyres have blazed in city parking lots, and bodies have washed up on the banks of the holy river Ganges, having been immersed by relatives whose villages were stripped bare of the wood needed for cremations.
Lacking beds, drugs and oxygen, many hospitals in the world's second-most populous nation have been forced to turn away droves of sufferers, while tales of desperate relatives searching for someone to treat dying loved ones have become sickeningly commonplace.
Although the infection curve may be showing early signs of flattening, new cases are likely to fall off slowly, according to virologist Shahid Jameel.
"We seem to be plateauing around 400,000 cases a day," the Indian Express newspaper quoted him as saying. "It is still too early to say whether we have reached the peak."
Indians need vaccines "here and now", the chief minister of West Bengal state, Mamata Banerjee, said in a letter to Modi. India has fully vaccinated barely 2.5% of the population.
Delhi had run out of its reserves of shots and had to close down several centres, deputy chief minister Manish Sisodia told reporters.
India is using the AstraZeneca vaccine made at the Serum Institute in the western city of Pune and Covaxin by Bharat Biotech but production is well short of the millions of doses required.
The country accounts for half of COVID-19 cases and 30% of deaths worldwide, the World Health Organization said in its latest weekly report.
The full impact of the B.1.617 variant found in India, which the WHO has designated as being of global concern, is not yet clear, it added.
British Prime Minister Boris Johnson said his government was looking at all possible solutions to tackle a surge in cases of the coronavirus variant first detected in India, including in the northern English town of Bolton.
"It may be more transmissible ... maybe even considerably more transmissible," he told parliament.
Daily infections are shooting up in the Indian countryside in comparison to big towns, where they have slowed after last month's surge, experts say.
More than half the cases this week in the western state of Maharashtra were in rural areas, up from a third a month ago. That share is nearly two-thirds in the most populous, and mainly rural, state of Uttar Pradesh, government data showed.
Television showed images of people weeping over the bodies of loved ones in ramshackle rural hospitals while others camped in wards tending to the sick.
A pregnant woman was taking care of her husband who had breathing difficulties in a hospital in Bhagalpur in the eastern state of Bihar, which is seeing a case surge its health system could barely have handled at the best of times.
"There is no doctor here, she sleeps the whole night here, taking care of her husband," the woman's brother told India Today television.
In a corridor outside, two sons were wailing over the body of their father, saying repeatedly that he could have been saved if only he had been given a bed in an intensive care unit.
At the general hospital in Bijnor, a town in northern Uttar Pradesh, a woman lay in a cot next to a garbage can and medical waste.
"How can someone get treated if the situation is like this" asked her son, Sudesh Tyagi. "It is a hell out here."
(Reporting by Anuron Kumar Mitra and Manas Mishra in Bengaluru, Shilpa Jamkhandikar and Aishwarya Nair in Mumbai, Tanvi Mehta in New Delhi, Subrata Nagchoudhary in Kolkata and Stephanie Nebehay in Geneva; Writing by Raju Gopalakrishnan; Editing by Simon Cameron-Moore, Clarence Fernandez, Mark Heinrich and Giles Elgood) ((Lincoln.Feast@thomsonreuters.com; +61 2 9373 1826; Reuters Messaging: email@example.com))