Police in the state of Uttar Pradesh said the user had been spreading false rumours and generating "fear" online.
Police in India have pressed charges against a man who tweeted asking for oxygen.
Amid soaring COVID-19 cases, Indians have been using social media to appeal for oxygen supplies, medical equipment and hospital beds.
One user on Twitter posted he was in need of an "oxygen cylinder as soon as possible," contacting a Bollywood actor.
But Indian police in the state of Uttar Pradesh said the user had been spreading false rumours and generating "fear" online.
The user's tweet did not make any reference to COVID-19, nor mention that he or a relative was suffering from a disease.
"When contacted immediately, it was learned that [the man's] cousin's maternal grandfather was 88 years old ... he died of a heart attack at 20:00 (local time)," Amethi police tweeted. The exact circumstances around the man's death remain unclear.
"He neither had COVID nor was there any medical consultation of oxygen," police said.
"At this time it is not only condemnable, but also a legal offence to post such type of fear-generating posts on social media."
Police added that the man's widely-shared tweet had inspired others to make allegations against the government of Indian Prime Minister Narendra Modi.
In a later video, the authorities confirmed that they had begun legal action and appealed for citizens not to share "misleading facts and rumours" on social media platforms.
Opponents of PM Modi said the police charges are the latest indication of eroding freedoms in India, described as the world's largest democracy.
Earlier this week, Twitter removed dozens of tweets criticising India's handling of the COVID-19 pandemic, at the request of the government.
The social network said are committed to promoting healthy conversations on their service and tackling misinformation during the COVID-19 pandemic.
Facebook has also temporarily hidden posts calling for the resignation of Modi under #ResignModi, before later admitting to Euronews that this was a mistake.
"We temporarily blocked this hashtag by mistake, not because the Indian government asked us to, and have since restored it," a Facebook company spokesperson said.
On Thursday, India set another global record in new virus cases, with more than 379,000 new infections. The Health Ministry also reported 3,645 deaths in the last 24 hours.
The country has now reported a total of over 18.3 million COVID-19 cases, and images across the country show many state healthcare systems are being overwhelmed.