By the same you can sue anyone referencing your articles in a research paper.
Technically, you are supposed to get permission before citing a copyrighted source, even in a non-profit or academic setting. An author or copyright holder can sue a researcher for copyright violations if they find fault with the use.
There is a
"Fair Use" clause in US copyright law, and that allows "limited" use of copyrighted source, but there's no strict definition on what "Limited" means - just whatever you can justify in front of a judge and convince them of, if you get called to task.
Generally, unless someone is making money of someone else's copyright (which is the case here), or someone is using Fair Use to tarnish a brand or reputation - there's no real money in going after infringement, so you don't see a lot of enforcement. Unless you are Nintendo, then you just go after everyone.
If the NYT can prove that ChatGPT was intentionally and specifically trained using their material, or if they can catch ChatGPT using exact quotes or citations from NYT material without permission, they have a strong case and would probably at least settle something, if not outright win.
If it's just a case of ChatGPT scraping everything up in the web, and NYT had a lot of stuff hanging out there in the public domain - that's a less strong case, but has much wider implications. There are a lot of copyright holders out there with stuff hanging out, and it goes to show if scraping for commercial use is covered under Fair Use. It could impact non-AI things like search engines, which rely (in part) on crawling and scraping to generate metadata and indexes.