Can I travel with cremated remains (father's ashes) from USA to India?

I have been researching the transport as well.

The most comprehensive list of requirements I have found at this webpage:
https://ihpoe.mohfw.gov.in/humanremains.php

I did a quick google search for "health officer Delhi Airport" and found the telephone and email contact information for the health officer. It appears the 48 hour notice can be satisfied by sending an email to the health officer.

I would like to share my last year experience from San Francisco to New Delhi on Air India direct flight. My wife passed away last year and I also carried the cremated remains in my hand luggage.

I needed to get attested documents from Consulate General Of India San Francisco.
This is the website: Consulate General Of India San Francisco

I just followed all the requirements and good news was I applied through mail, didn’t need to go in person although you can apply in person as well. It took only 4 days to get all the supporting documents from the Indian Consulate.

Finally, we carried cremated remains in our hand luggage. So basically depending on your residence in USA, you may be able to get all your documents through courier.

Another option is you can cargo the cremated remains through different airlines as well.

Can we carry cremated remain ashes while we travel to India? what are
the rules and regulations we must follow?

Yes, you can carry and travel with cremated remains from USA to India. You are supposed to meet TSA requirement, Airline policy and Indian embassy/consulate requirement.

Tsa Regulations

From the My TSA, you can carry cremated remains and there is an advice to check with your airline if you are allowed cremated remains as checked baggage or carry on bag. Further cremated remains should be transported in a lightweight container made of wood or plastic. please see the following guideline:

“Under no circumstances will a Transportation Security Officer (TSO)
open a crematory container. To facilitate screening, we suggest that
you purchase a temporary or permanent crematory container made of a
lighter weight material, such as wood or plastic. If the container is
made of a material that generates an opaque image, the TSO will not be
able to clearly determine what is inside the container and the
container will not be permitted.

Even if an item is generally permitted, it may be subject to
additional screening or not allowed through the checkpoint if it
triggers an alarm during the screening process, appears to have been
tampered with, or poses other security concerns. The final decision
rests with TSA on whether to allow any items on the plane.”

Airline Policy

Cremated human remains from Qatar Airways:

Cremated human remains shall be accepted for carriage as checked
baggage, provided that the passenger is in possession of all necessary
documentation, including a certified copy of a death certificate, and
the casket or urn containing the cremated remains is packed in a
sealed outer box or case.

Cremated human remains can be accepted for carriage in the passenger
cabin provided the state of origin does not classify it as restricted.

Indian Embassy/Consulate Requirement

From the official Consulate General Of India, you must meet all the following requirements:

Carrying of ashes of the deceased to India

Airport Health Organisation under Directorate General of Health Services, Ministry of Health & Family Welfare, Government of India has notified that for obtaining public health clearance of dead bodies and cremated remains transported by international airlines at Indian airports, as per the The Aircraft (Public Health) Rules, 1954, the following needs to be observed:

• The consignee as well as the air transport service shall give to the Health Officer of the airport of arrival advance intimation of
  at least 48 hours, of the importation of the dead body or human
  remains or ashes of the cremated bodies.
• There should be a proper death certificate issued by a Medical Officer in English or translated into English in respect of the dead body. As per International health regulations of WHO, Death Certificate should be in English or French only. The translation of death certificate sometimes causes undue delay and frustration among the attendants.
• The death certificate showing cause of death as pending or mentioned as cause of death to be ascertained after postmortem or
  cause of death unknown or as natural death shall render the Death Certificate as invalid and the dead body/human remain shall not be granted NOC by Health Officer of the airport. Embalming Certificate should be enclosed.
• The NOC should be obtained from Indian Embassy/Consulate of the country from where body is transported.
• As per the Provisions of The Aircraft (Public Health) Rules, 1954 Part IV Rule No.41 no person shall bring into India any dead-body or human remains of persons who may have died of yellow fever, plague, anthrax, glanders or any other diseases as may be notified by the Government of India for this purpose. Provided that nothing in this rule shall apply to properly cremated ashes of dead bodies or human remains.

A google search Human Ash Qatar Airways
unearths this first result

They can carry humain remains as shipment and/or checked luggage and/or cabin luggage, depending upon type of remains, and will need a copy of death certificate, NOC from Indian Embassy in USA, Cremation Certificate (or embalming/coffin/relevant certificate), accompanying passenger bookings and few others.

Relevant Quote from above Qatar Airways Page:

Cremated human remains

Cremated human remains shall be accepted for carriage as checked baggage, provided that the passenger is in possession of all necessary documentation, including a certified copy of a death certificate, and the casket or urn containing the cremated remains is packed in a sealed outer box or case.

Cremated human remains can be accepted for carriage in the passenger cabin provided the state of origin does not classify it as restricted.