Not really, we owe our democracy and civilization to an 18th century movement called the Age of Enlightenment. Those philosophers developed an idealized fantasy of what greek culture was supposed to have been derived from very old books. Only the Roman Empire had direct contact with ancient Greeks and took over some of their gods and culture, but that was lost when the empire fell. What we did learn from the old Greeks however was mathematics and science, which was arguably much more important than any cultural impact.
Ironically Europe had very much to do with naming really everything on earth, including America. If we wanted we could have sticked with calling you India. North India and South India. The United States of India. U.S.I.