My kicks was super dirty when I bought it, as it was on display where passing cars can kick up dust. When I took it back for my pro pack visit, a couple days later, they did a full exterior detail. They "detailed" the interior, but forgot the windows :/
If you have to take your car back for a pro pack install, they will detail it at the same time.
By "they" do you mean the dealership? Each time I've purchased a vehicle, the dealership cleans it up before handing over the keys. If the dealership didn't clean it up, give them a call, tell them about the stain, and see if you can take it in for a cleaning.
I followed your advise and went back to have the car detailed which they did. However, there are still a lot of water spots and stains all around the car as shown in the pictures below. Not sure if this can still be removed as they may have seeped through the paint.
Are those the same spots that were there previously? I ask because the water that people use (ie not rain water) has mineral deposits and will very often leave water spots like you've shown if left to air dry. The bad news is that the deposits could get "baked in" but chances are good that they will also wash off.
My significant other is the resident car detailer at our house so I can't get into better detail than that. I know he likes a brand of products called The Chemical Guys and he always dries our cars with microfiber towels to prevent the water spots. If washing your car by hand isn't your thing see if there's a business nearby that offers the service. You don't need to have it "detailed," but rather washed and dried. (An actual detailing service is expense.) By the way, if the dealership left these spots behind and didn't at least discuss them with you, they did not detail the outside of your car, they just washed it.