I think he means communal spaces in densely populated urban areas, not national parks and forests that are remote enough to keep most of the antisocial scum away.
Take for instance, Pioneer Square in Seattle. If you look at the structure of the parks, buildings, sidewalks and streets it should be very nice. Lots of nice trees, wide sidewalks and it's pretty flat and easy to walk around. But in reality it's filled with awful people which ruins it.
we have public open spaces as large as some EU nations
feel free to tally up parkland in North America by acreage vs the EU...