Firstly: It is established through the Sunnah that the Prophet Muhammad (peace be upon him) performed this act. It has narrated that Ubaidullah bin Abi Rafi reported from his father that the Prophet (peace be upon him) called the Adhan in the ear of Al-Hasan when Fatimah gave birth to him. [Reported by Abu Dawood, At-Tirmidhi, and Ahmad].
Secondly: There is no established hadith in the Sunnah to my knowledge regarding reciting the Iqamah in the left ear of the newborn, even though it has been reported from some of the Tabi’un (successors) that they practiced it. It has been narrated that Umar ibn Abdul Aziz used to call the Adhan in the right ear and recite the Iqamah in the left ear when a newborn was born to him.
Thirdly: Scholars of the different schools of thought have differed regarding reciting the Adhan and Iqamah in the ears of a newborn. The majority view among the Hanafis, Shafi’is, and Hanbalis is that it is recommended (mustahabb), while some scholars among the Malikis consider it disliked (makruh).
As for what has been mentioned regarding the reason the prayer, there is no evidence for it, and it does not resemble the statements of scholars. I have not come across anyone who has mentioned this in what I have read.
Similarly, it can be countered that there are multiple prayers that are performed without the Adhan or Iqamah, aside from the funeral prayer. Examples include the Eid prayers, the eclipse prayers (both solar and lunar) according to those who perform them, the rain prayer (Salat al-Istisqaa) according to those who observe it, and the Taraweeh prayers.
And Allah knows best.
Fatwa by Dr. Khālid Naṣr