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Considering you are a Greece passport holder you can simply hop on a plane and go to Chile for a stay of up to 90 days. Greece nationals do not need a visa if they intend to visit Chile for less than 90 days.
Kindly make sure your passport is at least 90 days till expiry, prefer to be 6 months.
resourceful links:
https://chile.visahq.com/requirements/Greece/resident-United_States/
https://en.wikipedia.org/wiki/Visa_requirements_for_Greek_citizens
If you want to clear any confusions you can call the embassy @202-785-1746