We call them Farmer's Markets in the US. You might not find them everywhere, but I'm from one of the more rural parts of the country, and they are easy to find where I live. You can buy fresh fruits and vegetable at Walmart, and you can find Walmart all over.

We also call the shop "the greencrocer's" in Australia, but we also say "the fruit & veg (shop)". (Soft "g" for "veg".)

It depends where you are. In smaller towns and some suburbs we still do have a few dedicated greengrocer's shops, just as we have a few dedicated butcher's, baker's, and fishmonger's shops. Unfortunately, they can't compete on prices with the big supermarkets, and are disappearing fast.

In the US it's either supermarkets or fruit and vegetable stores that don't have particulcar names. I've lived mostly in Queens, NY and there are actually quite a few of those. We either call them by itheir names ("let's get some fruit at the Green Market") or by some other name like "pick up some fresh salad at the Greek guy's place" or "I got these apple from the fruit place on the corner".

In suburbia New Jersey we have supermarkets. But a store dedicated to fruit and vegetables... yes, but they are usually a farm selling it's seasonal produce.