We do have those "national parks" here as well, of course we can't compete in size as we don't have the landmasses for that.
But it's as you say, you do need a car if you are living in a City.
However, here in Liverpool I am somewhat fortunate, I can board a regional train that takes me to Snowdonia or the Lakes and a bit further to Scotland and there is still bucket loads of "untouched" nature.
Untouched in quotes as you hardly find any untouched nature here in Europe. Most of what we think as 'nature' is man-made as well.