"I AM DADA to the cutest baby girl," he wrote, using the Hindi word for paternal grandfather.
The 38-year-old L'Oreal spokeswoman is married to actor Abhishek Bachchan.
This is the couple's first child.
India's boisterous news channels were unusually respectful of the family's privacy after an industry group issued new guidelines instructing television journalists to cover the birth announcement with restraint. According to news reports, the Broadcast Editors Association issued the guidelines last week.
The new rules included no pre-coverage of the birth and no outdoor broadcast vans outside the hospital. The event was to be reported only after a formal announcement by the Bachchan family.