Don't travel without passport
Please try to get the passport with a letter from the hospital and a copy of the application form. I tried going directly to the county office and got the Birth certificate within half hour. Since we had a family emergency we got the passport within 2 hours and Indian Visa within 2 hours all on one day.
However DO NOT TRAVEL without a passport. My friend took her daughter without Indian visa, but they did have the passport, at the Singapore airport, the airport authorities asked them to leave the child behind. They had to wait in Singapore, get a interim Indian visa and then leave for India. And the the Indian visa was only for 15 days, hence they had to apply for extension again.
With passport only they had so many issues, think about no passport and no birth certificate, they might question about the proof that the baby was born in US I guess.
My advice is get the passport and the visa.