You called her a karen five times in the space of this post. That framing doesn’t help you. You don’t know that she’s an innately unreasonable person, you just know that she’s unreasonable in a specific moment where several people have failed her in a serious matter.
does it really help to simply stay next to my colleague, letting her do the talking while I do nothing but looking at the karen in the eye?
Staying quietly next to your colleague does help your colleague but this sustained eye contact sounds like an escalation. Try to express condolence instead of contempt.
what if, each time the karen opens her mouth I repeat ‘calm down’ ad nauseam till she either tires, shuts up or walks away?
That is an escalation of hostility. Any time you find yourself thinking “maybe this will shut them up”, that’s you looking to defeat an adversary. The goal here is to make her feel better, not even more hopeless.
what do you say or do to support your coworkers while they’re being verbally abused that somewhat defuses the situation?
Redirect. Gently point out that the coworker isn’t to blame for the problem and suggest focusing on how to proceed from here.
what if avoiding conflict is a trait of mine to the point that I let people walk all over me?
how do you resist the urge to walk away? Situations like this trigger my fight or flight response.
All there really is to do is practice.
what if I have to do this with a man and it gets physical? If somebody strikes me and I strike back, and I can guarantee you I’m striking back, I’m as guilty as the first aggressor.
This is one of the reasons it’s nice to have coworkers standing by your side. If it does get violent, a third party can split that up easier than a participant.
Broad strokes, make it clear that you would also prefer that her father get his scheduled treatment.
You asked how to defuse the tension, I’m telling you how to defuse the tension. If that’s not what you want to do then I can’t help you.
EDIT: That was kinda glib on my part so I’ll expand some more.
What you’re asking me to do right now is acknowledge the legitimacy of your grievance. And I do, of course. Nobody deserves to be abused.
The goal here is to make the abuse stop happening, though, right? So the best course of action is the course of action that stops it. If you discount the possibility that she can behave reasonably, there’s no possibility of talking her down from her current state of aggression. Essentially, you both want the same thing: not to feel upset, not to feel hurt. If you can make her stop feeling that, she will stop causing you to feel that. It’s not fair but you didn’t ask for fairness. You asked for how you can produce results.
She has framed this as her on one side versus you (you being a proxy for the hospital) on the other. Option A is you accept her framing and remain in conflict and both of you continue to cause each other pain, option B is you influence her to change her framing into one side being both you and her and the other side being the situation. If you succeed at option B, both of you stop causing each other pain. So you acknowledge the legitimacy of her grievance. Her father was supposed to get an operation that day and he didn’t. Nobody deserves to have their medical treatment withheld, even temporarily, even if it was an elective procedure.