Call local emergency number using a foreign mobile number

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.

In the United States, you can make an emergency 911 phone call from a mobile phone without a SIM. As long as your phone can communicate with the network (supports the same frequency and standard), you can call 911. This is governed by the Federal Communications Commission (FCC).

The FCC's basic 911 rules require wireless service providers to transmit all 911 calls to a PSAP, regardless of whether the caller subscribes to the provider's service or not.

If you do not have service, there are even instructions about calling back in the event your emergency call is disconnected:

If you do not have a contract for service with a service provider and your emergency wireless call gets disconnected, you must call the emergency operator back because the operator will not have your telephone number and cannot contact you.

Data point:

In NZ the emergency number is 111.

112 works to call emergency services from a GSM phone - I just tried.
Nobody has ever told me this before now.

Dialling 112 from a landline accesses a very slowly and clearly spoken message - ~~=

• "The NZ emergency service number is one one one.
  That is ONE ..... ONE .... ONE.
  Please hang up the receiver then slowly and carefully dial one one one"
  Repeats ....

In any ITU country, dialing 911, or 112 should connect you to emergency services if the phone physically works. (I probably shouldn't go and test this, but any emergency number I could think of worked where I live a few years ago.)

112 is more standard in the EU, and 911 in North America. I have gotten connected to emergency services in the EU dialing 911, but don't count on it.

It's separate from your phone plan, and separate from regular phone number routing. So it isn't like a regular phone number.

Per Emergency Communication Version 1.0 01 July 2021 - GSMA (PDF)

When an end user dials a number related to an emergency, or a [sic], the UE shall check if this number is identified as a valid emergency number. The following nominal cases are identified by the UE as valid emergency scenarios: [...]

• Standard emergency numbers dialled by the user (112 and 911) [...]

If the UE has identified an emergency number (as defined above), the UE initiates a emergency call setup procedure, enabling high priority in case of network congestion.

Translating from telecomese to English, if any GSM-compatible (read: modern) "User Equipment" (read: mobile phone) anywhere in the world dials 112 or 911, the network must treat it as an emergency call and route to emergency services. All other numbers are country-specific.

Obligatory disclaimer: just because the official standard says something doesn't mean every phone or every network actually complies in real life, but in practice this is basic stuff and they all do. Also, the relationship between GSM the original 2G standard, GSMA the association, and newer standards like 3/4/5G is complex, but the doc above was last updated in July 2021 so yes, it's still current and relevant.