What is the ruling of Islam regarding a midwife in a non-Muslim country who performs the Adhan in a newborn's ear, hoping that this will have a future influence on the child to embrace Islam?

There is no objection to a woman working as a midwife, as this is a role specific to women. As for the adhan in the ear of a newborn, it is a Sunnah according to the majority of scholars for the guardian. The Prophet, peace and blessings be upon him, gave the adhan in the ear of al-Hasan. Umar ibn Abd al-Aziz would give the adhan in the right ear and the iqamah in the left, and he would say the tasmiyah (Bismillah). The more likely view is that it is not prescribed for the children of disbelievers, and it is a condition that the one performing it be male, and that the newborn be a Muslim.