Will Apple Watch Series 7 bought in USA work with GSM network in India?

There are some carriers in India that will ignore that, but you should check with your carrier before you buy. The 900MHz frequency band will be missing in a U.S. purchased Apple Watch, so if the carrier only uses 900MHz in some area, it will be a dead zone to you. But if your iPhone is with you, then the Apple Watch will piggy back off the iPhone.

Note: your Apple Warranty will not apply, as it is country specific, unless you are returning to the U.S. frequently.

A cellular watch bought in the US will only function in North America,Canada,Mexico and Puerto Rico anywhere else it will function as a GPS only watch.

If it's a GPS only watch then it will work just fine.