We realized that my child's passport expired and we are travelling tomorrow from the US to India. She has a US green card, and an Indian passport. Will she be able to travel?
Unfortunately your daughter cannot travel.
Even if you were to board the airline - which is doubtful (see the Timatic note) your arrival and transit will be an issue since often passports are checked during transit changes.
As per Timatic (the database airlines use to check travel requirements), it states plainly (I highlighted the relevant part):
Passport Exemptions: Nationals of India with an Emergency Identity Certificate. Nationals of India with a photo identity card, if arriving directly from Nepal.
Document Validity: Passports issued to nationals of India must be valid on arrival.
Additional Information: Indian law does not permit dual nationality for nationals of India. An Indian national holding dual nationality must contact their embassy/high commission for further information.
Passengers arriving from Guinea, Liberia, Mali and Sierra Leone at Bangalore (BLR), Chennai (MAA) ,Cochin (COK), Delhi (DEL), Hyderabad (HYD), Kolkata (CCU) or Mumbai (BOM) are subject to enhanced passenger screening and medical checks.
Nationals of India, traveling directly from Nepal, aged:
up to/incl. 9 years of age, do not require any documentation, if accompanied by parents/guardian; or
10 up to/incl. 18 years of age, can travel with a certified identity document containing photo, issued by the Principal or Head of an educational institution.
Nationals of India with a passport that contains a stamp 'extended by the issue of fresh booklet' must travel with the new booklet.
One can make an argument that citizens cannot be denied entry, and there is truth to that statement - but your problem is a bit basic, that is, your daughter won't be allowed to board.
My recommendation is to apply for an emergency travel document immediately.