Actually, ignoring for now the issue of whether someone poses a risk to others.
IF a docyor considers you are suicidal, then you pose a risk to YOURSELF and I would imagine that gives them a duty to you - their patient - to do something about it. The fact that none of the doctors involved sought anything more as an action than a referral or a sick note or whatever suggests that none saw anything that escpecially worried them.
I don't think they need any kind of "duty to inform an airline" - the pilot's a threat to himself anyway, and is that not enough?
With the full acknowledgement that i don't know what any particular law might say regarding doctors and potentially suicidal patients.