What is courtship in Greece like? Is it very similar to American culture?

I'm at the early stages of dating a Greek man who has recently moved to the United States (a couple of months) and I know that there are many cultural differences... so I'd like to know how different are the male and female roles during courtship... Thanks!

This is a real problem. You see, Greece has been long the crossroads between East and West and Greek people have different cultures. All Greeks respect family, religion and high education.
However, if he's brought up in the old fashioned matter, it would not be a good idea to speak openly about flaws in the characters of his close family persons. He's something like Guss of the Greek family in "My big fat family". He believes that he is the master of the family (but in reality it's his wife) and he is rather difficult to communicate. Avoid talking religious matters or politics until you get to know him better and -it goes without saying- that you avoid speak badly about Greece despite the fact that he may speak very badly himself about his country.
If he's brought up in the Western way, things are much easier... He considers you to be equal and his behaviour tends more like the American one, but try to keep a rather low pace and let him take the initiative. What you can do, is to create the conditions to make him ask you what you want, but let HIM ask you. Aggressive women are welcome for bed but not for something more ...permanent.

I wish you good luck and contact me directly if you feel like it. I'm Greek and have the experience.

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ngiapapa say it like it is Good luck !