What is the medical term meaning study of the newborn?

neonatal
A newborn is a neonate: neo is from the greek Word neos, meaning new or recent and nate is from the Latin Word natus, which means 'to be born'.
Someone just born would be neonatal until they reached 28 days of age.
The next stage of development is infancy.
Neonatal describes the term just after a baby is born.
Neonatal means pertaining to a neonate, or newborn.