Your health insurance agency will give you list of hospitals they support in your city. There is no need to register with a hospital on your arrival. However, you can take a proactive approach and find a local doctor and identify a good hospital for emergency treatment as soon as you arrive and before you need to use them. Private medical care is readily available in India and is of a much higher standard than public care. A number of hospital chains offer high quality medical care, are well-staffed and provide clean facilities and modern equipment and can undertake most common procedures. By law, hospitals are required to attend to any emergencies that come to them.
To be admitted to a hospital for non-emergency purposes, you will have to be referred by a specialist or a doctor attached to that hospital. In some cases the doctor will arrange the admission for you–in most cases you will have to arrange it yourself. Make sure to carry your passport and any relevant medical records with you to the hospital. You may have to pay in advance (or at the time of release from the hospital) and then have it reimbursed by your insurance provider–this depends entirely on your policy. Many hospitals also have separate international patients cells catering to foreign patients and might offer services in the language you comfortably speak. All hospital staff you will interact with will speak fluent English.
In the case of an emergency, there is no need to call ahead and you will not be required to present any specific documentation. At some point, you will be required to show your passport and insurance card. Try to have them with you, but if you do not, it will not stop you from being treated. You will not be required to pay in advance or make a deposit. Cash, cheque, and credit cards are all accepted.