Seriously, is the race of a made up character from a modern fairytale such an issue?
Well, here's a question. If Ariel's black, presumably Poseidon's black, and all her sisters are, too. In fact, literally every merman/maid we see in the animated version will be black. So now it's a movie about a black girl who desperately wants to escape her country/life to go live in the "civilized", "modern", "better" world. It's not exactly the most uplifting interpretation of the story.
The HCA version was about unrequited love, unrealistic/impossible desires, trying to belong where you don't, and is a cautionary tales for children to be good (I think it ends with a sort of Flying Dutchman style "wander the world until she can find enough good children" thingie - but it's been a while and I may be confusing versions).
Do we rewrite the story
not to be about seeking a better world? To
not refer to (black) immigrants coming to the (white) civilized world of milk and honey?
On one hand, no, what color she has doesn't matter to the story - HCA describes her as pale but it's not an essential part of her character (like, say, Pocahontas...oh wait, we might as well make her blue and transport her off-world). OTOH, Aladdin or Mulan could be rewritten and transported to another place to be about white people without any problems. A thief using magic to impress a princess? Nothing Arabic about it. A girl pretending to be a boy to keep her family's honor? heck, I've
read that book set in medieval France before.
I'm not necessarily opposed to characters changing color (or gender, or sexuality, or hair length, or...) if they aren't essential to the character (Black Panther or Hitler, to take obvious examples). I do have to wonder about why this is such a thing now. Apparently Hollywood is so story-starved they can't think of
anything that might sell with black or female main characters that doesn't involve taking a story from a Western European or American folklore and changing it around a bit? Princess and the Frog showed some people at Disney manage to understand.