For all countries, you can just dial 112.
Dialling 112 will direct you to the same emergency call centre as phoning the country's emergency number would.
In the EU, emergency call centres must provide a translations service.
In some countries, a phone does not even need to have a SIM card present to dial 112.
112
(emergency telephone number) : 112 is a common emergency telephone number that can be dialed free of charge from most mobile telephones and, in some countries, fixed telephones in order to reach emergency services (ambulance, fire and rescue, police).
112 is a part of the GSM standard and all GSM-compatible telephone handsets are able to dial 112 even when locked or, in some countries, with no SIM card present.
It is also the common emergency number in India and in nearly all member states of the European Union as well as several other countries of Europe and the world.
112 is often available alongside other numbers traditionally used in the given country to access emergency services.
In some countries, calls to 112 are not connected directly but forwarded by the GSM network to local emergency numbers (e.g. 911 in North America or 000 in Australia).
Note: data-only sims might not be able to dial the emergency number in some countries.
travel.stackexchange.com/users/105447/Daniil
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.
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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.
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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 ....
travel.stackexchange.com/users/3301/Russell McMahon
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.
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I read about  How to make an international emergency call?  but 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.  Though I have mentioned USA, I would also like to understand how it works in other countries as well.
Call local emergency number using a foreign mobile number
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For all countries, you can just dial 112 to connect free of charge to emergency services. Dialling 911 should also re-route to emergency services anywhere. You can call 112 or 911 internationally.
You can dial 112 from any country. Calling 911 works in the USA and other countries.