Reference: I’laam al-Mu’aasireen bi-Fataawa Ibn ‘Uthaymeen – Page 358
Question: Is there any mention in the Sunnah that the adhaan and the iqaamah are pronounced in the ear of the newborn immediately after birth?
Response: [Indeed] it is mentioned in the Sunnah, however, the ahaadeeth regarding the iqaamah are weak, whilst the [ahaadeeth regarding the] adhaan are stronger. So because of this, we say the adhaan should be pronounced in the right ear [of the newborn] immediately after birth. Most people now[adays] are unable to do so because the nurses in the hospital are the ones who assist in childbirth, however, if a person manages to attend the birth, then it is befitting that he pronounces the adhaan in the newborn’s right ear – not in a loud voice, rather in a low voice so as not to harm the [newborn’s] ear. Some of the scholars have said – this is so that the first sound which he comes to hear is the adhaan and the call to prayer.
