I think Ariel didn't really know how to write, the quill Ursula used was enchanted and I figured it kinda wrote her name for her. partially. So that's why I don't think she wrote to Eric what was going on. Also, he might think she's crazy if she wrote...

Yeah. The quill from Ursula, just like King Triton's trident, were both enchanted objects, which could allow mermaids/mermen to write their names, but not whole sentences.

Occam's razor, aka the simplest explanation is the most likely one: Ariel didn't write anything down for Eric because it's a plot hole and also she's dumb.

Why would she know how to write only her name and nothing else? That's weird.
We can assume Ariel knows how to read - even that dinky little seahorse with the scroll can read - so why wouldn't she know how to write as well? She's privileged undersea royalty, I feel like it's a safe bet that she's been educated in something as basic as written communication.

Too many suspensions of disbelief needed to go with no, imo.

She can't write underwater I guess, can she?

What UnholyNoise said.

Agree with UnholyNoise. Plus, there obviously isn't a language barrier between merpeople and humans, since Ariel understood everything Eric said to her.

I agree that the quill was enchanted. She could quite possibly read (Ursula did produce a scroll with writing for her to read), but I doubt writing was ever important to the merpeople, they probably magically produced anything written.

I'm pretty sure they'd use squid ink or something to write things down. But as for her not writing to Eric about the whole situation there are two things I must say about that.

1. It's a freaking fairy-tale! It doesn't run on logic! It runs on emotion! It's the same as why didn't The Blue Fairy just make Pinocchio good and already know the life and moral lessons he was supposed to learn? Why couldn't Glinda just tell Dorothy she could go home with the slippers and that she wouldn't have believed her if she told her? How has the Wicked Witch gone her whole life without being in contact with water, like having a drink or taking a bath (Does she spell like sh*t?)? Why didn't the Evil Queen just make a potion to make herself more beautiful than Snow White if she can make herself ugly or just give Snow a potion to make her ugly? Why didn't Prince Charming describe what Cinderella looked like (Blue eyes and blonde hair) to narrow it down how many women could try on the slipper to find Cinderella? It's just the way fairy-tales work and that's part of their charm.

2. Even if Eric was interested in the stories of merpeople, it doesn't mean he believed them like the sailors did. Besides, a mute girl that is wearing rags and acts strange when you first meet her writing that she was the one who saved you, was a mermaid, sold her voice to a crazy sea witch to become a human, and he has to give her the kiss of true love in the next three days before the sun sets on the third day or she'll turn back into a mermaid and belong to the sea witch? Yeah, even if he believed in merpeople, I doubt he'd believe all of that because that sounds pretty far fetched. She'd end up getting locked up in the looney bin until she turned back into a mermaid and Ursula came to take her back.

I’ve always wondered this. I like to think she didn’t know what humans used to write with so maybe she thought they were behind in that kind of technology. Or she saw a quill from the human world but since it looked different then the one Ursula had, she didn’t question it.

I wish there was an in-between answer because it's really not THAT hard to believe, especially because anything unorthodox about Ariel or anyone in her family can be explained away by the fact that they're essentially from another world.