Fifty years ago there were many shops in the US selling primarily milk, with ice cream and perhaps cheese being secondary offerings. One of these shops would have been referred to as a "dairy" (even though the same term was used for an agricultural business which milked cows) or as a "dairy store". These have largely vanished, and the few that remain have likely been relabeled "creamery".
About the same time that the dairies were disappearing, so were the "ice cream parlors", although, except in tourist areas these were rarely stand-alone businesses to begin with but operated inside a drug store or "ten cent store" or some such. These "parlors" did not generally serve "packaged" ice cream but rather cones, shakes, sundaes, etc.
Starting about 50 years ago, various ice-cream-focused shops began popping up, variously called "ice cream shops", "ice cream bars", and a few others. These generally split the business between selling cones/shakes/sundaes and selling packaged ice cream, and often they branched out into burgers, etc. There are still the "soft serve" remnants of these, such as Dairy Queen, but most of those serving "regular" ice cream have passed away or converted to "regular" restaurants.
These days, in the US, the vast majority of packaged dairy products sold is sold in "supermarkets" along with other groceries, meats, vegetables, etc. What the rare stand-alone dairy store would call itself is anyone's guess -- it would no doubt be some catchy name.