Moving the database to the same region as your web server will certainly improve performance. The electrons travel along the wire, not at light speed, but very rapidly, the issue is the packets you send must be processed at several routers and firewalls as it passes through countries / ISPs. This matter is complicated by the fact that the trip must be made back and forth several times before any information is transmitted.
If you cannot move the RDS database to the same region, there is the option of setting up a caching server in the India region. If the AWS Service Elasticache is available in the India region, you can spin up a cluster (if it's not available you could spin up your own server running memcached to do the same thing) and that will make the long trip to the US and back, and store a copy of the data in memory where it can be quickly accessed by your web server. This will only improve common read requests and all write requests would still need to go to the US.
Another option might be to spin up another RDS in the India region and institute a multi-master solution, where the two databases will synchronize with each other. It's a little dicey with them being so far apart, but it ought to be fine as long as there are a low volume of requests. However, a multi-master database set-up is not a trivial thing to set-up and manage.
If you have the capability of moving the web server to the U.S. (depending on the type of content you're serving) you can set up content distribution with cloudfront in the India region and customers in India will be able to access your content with better latency (but customer requests doing writes will still have to make the long trip). Also, when moving servers to the U.S. be aware of the Patriot Act and what it may mean for your ability to ensure confidentiality of the data you're storing (though I'm guessing this isn't an issue if you have your Database in the U.S. already).