Your friend was feeling attacked, and you walking away made her feel abandoned and unsupported. She just wants to know that you have her back. Yes, standing beside her to help show a united front would do that some.

Usually to defuse a situation like that, I would try to understand what Karen was actually upset about, and let her know she’s being heard (which is, ultimately, what she probably wants), but also let her know she’s taking it out on someone who doesn’t deserve it. Saying “calm down” is just going to piss her off. Saying “Hey, I understand you’re upset, and I would be too, but the staff here has no ability to schedule your father for surgery when there aren’t any doctors, and you’re yelling at the wrong person” might.