In the modern age of connected world and social media, research outcomes that are of direct interest & relevance to society are increasingly being shared and disseminated in news sources and social media platforms. Some studies have found that social media mentions of research papers can be an early indicator of their impact. India, which is now among the top 10 knowledge producers in the world, has more than 900 Universities that contribute to its research output. This paper tries to analyze as to what proportion of research output from the 100 most productive Indian institutions gets social media coverage. It is found that, while average social media coverage for India is around 28.5%, the coverage varies between 5% to 60% for different institutions. It is also observed that research output from institutions in some specific disciplines (such as Medical Science and Biological Science) attract more social media coverage as compared to others. The possible impact of geographical location (in a metro city) of an institution on social media coverage of its research output is analyzed as well. The findings present useful insight about social media coverage of research output of Indian institutions, which may be a proxy for societal relevance of the research work, and also indicate that suitable mechanisms need to be designed to promote dissemination of research results from Indian institutions in popular social media platforms.