We have asked that our instructors avoid taking phone calls, taking unrelated facetime video chats, and eating lunch in front of customers (if possible).
So to sum it up, you asked them to work when they are on your payroll. That is not unreasonable, that would be normal.
This employee claims that our policies culturally discriminate against him.
That's nonsense on multiple levels. One, culture is neither in the civil rights act (US) nor in the EU directive. Even if you could discriminate based on culture, it would be legal in most countries I would expect a Turkish person to migrate and work in. Second, the things you listed have nothing to do with culture. There is no cultural requirement to eat food in front of others. There certainly is no requirement to be on the phone or Facetime privately. What culture would that be, affordable mobile phones are not even 20 years old! Even if it was meant to say ethnicity instead of culture, there is no region in Turkey (or probably even the world) where the above applies.
So there are two issues here. Your employee is unhappy and your employee is making legal, unbased threats.
You can negotiate their happiness level. Maybe they misunderstood the rules. Can all their problems be fixed by taking a scheduled break at noon, having lunch and making their phonecalls? Maybe they don't know that. Maybe they can work half days at first until all their movement paperwork is settled? I'm sure a solution can be found based on the rules you have set, they sounded reasonable.
The other problem might need more attention: why are they threatening you? Why not cooperate and ask others how they fit in their schedule? That person certainly is not the only one with a "life". You need to make sure that next time they ask for exceptions if they have a problem, not threaten you with legal wordings they don't (seem) to properly understand themselves. No matter how good an instructor the person might be, you need people to cooperate and work together.