If I buy the Apple Care + from the United States, would I be able to make use of it in India?

It might except if you take a USA iPhone to India you may find you cannot get it repaired at all because Apple restricts mobile device repair to the original country of service.

http://www.apple.com/legal/warranty/products/iphone-english.html

Note this statement from the iPhone and iPad warranty:

"IMPORTANT RESTRICTION FOR iPHONE AND iPAD SERVICE.

Apple may restrict warranty service for iPhone and iPad to the country where Apple or its Authorized Distributors originally sold the device."

The bottom line is that in effect you cannot actually get a phone you buy in the USA repaired in another country even if you do have Applecare on it. Apple often just replaces an entire device with an identical refurbished unit instead of repairing it and there aren't any replacement USA model iPhones in the Indian Apple repair places. AppleCare extends the warranty and the warranty says Apple "may" (read that also essentially almost always" refuse service in another country. If you want to get the iPhone repaired, you have to get it done in the USA.