It is proper English. It's called a declarative question:
"A declarative question is a yes-no question that has the form of a declarative sentence but is spoken with rising intonation at the end.
Declarative sentences are commonly used in informal speech to express surprise or ask for verification. The most likely response to a declarative question is agreement or confirmation."
The answer to the text would either be "yes, she left" or "no, she didn't leave."
I could have responded to your statement by saying, "it is, though?" and the answer would have been "yes, it is proper English."
No, I'm fairly confident it's still incorrect in this case. The protagonist isn't actually asking a question in that case. Look at chapter 25, page 3. He fully believes that she left by herself. More notably, he's texting with someone who wasn't there, and doesn't have any information. How could this be a yes/no question?
I'm pretty sure it's just a quirk of how Japanese speech differs from English. I've seen it very frequently in translated works, and virtually nowhere else. You'll see statements like:
FMC: "I know this outfit doesn't suit me..."
MC: "It looks great on you, though?"
It seems to me that when Japanese people disagree with a claim, they frequently phrase it as a rhetorical question. The character in the text message in question wasn't asking if she left early, he was stating that she left early in a way that contradicted the other character's question.