What should I put in the "LastName" section while booking a flight with United Airlines, when my Passport only has "FirstName". Can I use my "FirstName" in both the places?
For US Visa puporses, it looks like you need to use your single name as "last name" and FNU as first name. FNU means "first name unknow". See https://in.usembassy.gov/visas/nonimmigrant-visas/ FAQ section. It's likely that United will adopt the same process since they need to communicate booking data and Visa data to US immigration authorities. However, you should call and ask them to confirm.
However, it's likely that you will run into a lot of issues and if possible you should try to get your Indian passport fixed before you travel. https://redbus2us.com/surname-blank-in-passport-first-name-as-fnu-in-us-visa-issues-solution/