The information about 112 in another answer is correct. It is also true of 911, so to address your concerns explicitly,
not sure if I will be able to call the local emergency number (USA - 911) using a foreign (India) mobile number when I am in USA.
You will be able to do that. If your phone does not recognize 911 as an emergency number, it sends the call to the local cellular network tower, which will recognize that it is an emergency call and send it to the emergency call center. In this case, it doesn't matter where your phone or phone number is from, because the call is handled by the equipment of the local telephone company.
Though I have mentioned USA, I would also like to understand how it works in other countries as well.
The same thing happens. Whatever provisions may be in place for recognizing certain foreign emergency numbers as such, the local network will always recognize the local emergency number.