Oh nursing is rife for this sort of bullshit.
It's part of the culture.
Management will only accept notification of sickness from the staff member themselves, irrespective of whether they are vomiting or delirious or semi-conscious. Then management bollock the staff member because they haven't given enough notice and will leave them short, or they bully staff with threats of sickness interviews and insinuate that the sick staff is telling porkies and a disciplinary will result if this is discovered 
It is one of the reasons that sick staff in hospitals often drag themselves in, because they hate leaving their fellow staff members short but also because it is just so awful to feel sick and hope for some understanding and compassion from your employer but instead get treated like a piece of shit on their shoe and have it implied that you are feckless and have let them down.
I will never forget being sick when I was employed as a lowly staff nurse on an Intensive Care Unit and I started feeling sick the day before I was rostered to work. Finally at about 2pm I rang my manager to inform them that I wouldn't be able to make it in for my shift that commenced at 7.30am the next day and got BOLLOCKED for not having had the decency to have called her that morning as now she would struggle to find staff to cover me
I also, like the OP, had had very limited time off due to sickness so manager's annoyance was definitely not due to a persistent pattern of work avoidance on my part.
OP, I agree that the nurse you spoke to was rude. However there is probably not much that you, or your daughter can do about that. However, if it is true [and there are witnesses to the fact] that she told staff that she believed your daughter was lying, this needs to be brought to the attention of management [by your daughter] and this is absolutely not acceptable.
And personally I can see no issue with your ringing your daughter's workplace to inform them that she was unable to attend her shift in 50 minutes time. It sounds as if the vomiting came on very suddenly and took your daughter by surprise. If you were available to call for her, why the hell shouldn't you? Surely that is just common sense rather than your daughter calling when her head is down the toilet bowl?
And then your daughter could have followed up with a call to management herself when she was feeling more able to do so.
It's really not a comparable situation to those on here who are saying well when I lived by myself I had no-one to call in sick for me so OP's daughter should also have called in for herself. What a ridiculous argument 