Is it necessary to treat stye (in eye lid) of 6 years old girl with antibiotics ointment or if it starts getting better with only warm compress then there is no need for antibiotics ?

Least invasive first: Because of rising antibiotic resistance rates, it is always a better approach to use the most conservative treatment first. Gently wiping/massaging eyelid, and applying warm compresses is the first level of treatment. If the patient responds to this with a decrease of symptoms and swelling, then use of antibiotics has been avoided.