This reminds me a lot of AI-based self-driving cars, which have gained quite a lot of trust already, but it also still has a long way to go. AI also makes decisions in many other places (e.g. fraud detection). I'd say ChatGPT is far behind that in terms of trust in anything resembling a regulated environment, but it might show a possible path of where such models might go in future, if trust increases.
Justifiable trust in a method or source of truth seems to come from 2 places:
Reliable results
If following the same method consistently produces results we've verified as true, this would increase trust.
ChatGPT says a lot that's right, but has frequently shown to just make stuff up, so that's not particularly reliable.
You could also question the types of reliability. If someone comes in with a common cold, you don't really want to diagnose them with cancer, or vice versa. You wouldn't really expect a competent doctor to make such mistakes, but rather to make mistakes when there's more ambiguity and uncertainty, where multiple conditions may meet the symptoms (which then arguably wouldn't be a mistake), or they'd miss something, or it's some rare disease. The mistakes of ChatGPT might be more arbitrary.
Understanding
Understanding how a method works and why and how individual outputs are generated could also increase trust.
As far as ChatGPT is concerned, we understand the low-level maths, and we understand what it does on a high level, but we can't explain why a individual outputs are generated (beyond "sticking this input into these equations give this output").
Unless something significant changes here, we'd probably need to rely primarily on reliable results, rather than understanding.
If anything, the understanding we have might decrease our trust, as we know that it just chains words together, based on what it's seen in the past, without much concern for nor ability to evaluate whether the result makes sense.
Side note: explainability is a big topic in AI, although the more complex models tend to fail there.
Also, the medical domain has a particularly high threshold for trust, as wrong decisions could very directly lead to people's deaths, and it doesn't have the benefit of, say, engineering, where every individual design can be reviewed and extensively tested before anyone's lives depend on it working (never mind failsafes).
Other domains may very well meet the threshold for trust before the medical domain would.
I could also see something like ChatGPT being used or supervised by a doctor, to aid diagnosis more than to diagnose by itself. It might also be used (mostly by private individuals, probably) in cases where doctors aren't available (although something manually curated, like WebMD, is probably better for that, even though that is not without problems).