Did I make a mistake? Is there any way to start using cellular feature in India with US watch model?
Yes, you made a mistake I'm afraid. The Americas-regional model of the Apple Watch 4 won't work in India. For that you'll need the Europa/Asia/Pacific-regional model.
In addition to that you'll need a supported carrier in order to get an eSIM for the watch. In India this is only supported with Airtel and Jio. For details refer to Apple Watch Carriers webpage and search for India: