Micah’s name day is August 14, the feast of Micah the Prophet.
Churches that follow the Old (Julian) Calendar keep the same feast on August 27 of the civil calendar. See everything commemorated on August 14.
The saint behind the name

Micah the Prophet
August 14, Prophet, Μιχαίας
The Holy Prophet Micah was one of the twelve minor prophets, prophesying in the kingdom of Judah in the eighth century before Christ, in the days of the Prophet Isaiah. He rebuked injustice and false worship, foretold judgment upon Jerusalem, and proclaimed the mercy of God who pardons iniquity.
Also celebrating on August 14
These names share the feast, many of them other forms of the same name:
What to say on a name day
The Greek wish is Χρόνια Πολλά (Chronia Polla), “many years.” A call or a short message on the day is what people remember. See name day wishes you can send and what a name day is.