What is the ruling on reciting Al-Fatihah immediately after the Adhan?

It is prescribed when hearing the adhan (call to prayer) for the listener to repeat what the mu'adhin (caller) says, then invoke blessings upon the Prophet, peace and blessings be upon him, and ask Allah for al-wasila (a high station) for him, based on the Prophet's saying, peace and blessings be upon him: "When you hear the mu'adhin, say what he says, then invoke blessings upon me, for whoever invokes blessings upon me once, Allah will invoke blessings upon him ten times. Then ask Allah for al-wasila for me, for it is a station in Paradise that will only be granted to one of Allah's servants, and I hope that I am he. So whoever asks for al-wasila for me, my intercession will be permissible for him."

It is also sunnah (recommended) to say after the adhan: "O Allah, Lord of this perfect call and the established prayer, grant Muhammad al-wasila and al-fadila (pre-eminence), and raise him to the praised station that You promised him," based on the Prophet's saying, peace and blessings be upon him: "Whoever says when he hears the call: 'O Allah, Lord of this perfect call and the established prayer, grant Muhammad al-wasila and al-fadila, and raise him to the praised station that You promised him,' my intercession will be permissible for him on the Day of Resurrection."

It is also sunnah to say upon hearing the adhan: "I bear witness that there is no god but Allah, alone, without partner, and that Muhammad is His servant and His messenger. I am pleased with Allah as my Lord, with Muhammad as my messenger, and with Islam as my religion." It is preferred to say this when the mu'adhin says "La ilaha illa Allah" (There is no god but Allah) at the end of the adhan.

As for reciting Al-Fatiha after the adhan, there is no basis for it in Islamic law.