YANBU. Having been in a similar situation (a month in hospital with a seriously ill baby) I can empathise. Those words about it being trivial compared to what was going on in your life are so true.
It's good that you have other supportive friends and family and it seems your relationship with your father is strong, as my guess is you really don't need this narcissistic woman in your life. Put your wellbeing and that of your baby first.
There's no way you should feel that you have to apologise. And, indeed, if you were to do so, as you say, it's hard to see how things could go on as if nothing had happened.
On the other hand, if you were to tell her, possibly in writing, how you were preoccupied with bigger issues, how you felt/feel as a result of her behaviour whilst acknowledging possibly that you shouldn't have hung up on her, or now regret doing so, could it be a way forward?
To be honest, even if you go down this route, there is no guarantee that she would "get it." Like other posters, I suspect, that this is some weird type of power game, or she is too self-absorbed to appreciate what you are going through. But it could just clear the air, and you would be showing yourself as willing to take on board her feelings, whilst making it known that she should respect yours, and make life easier for your dad and possibly other family members.
Perhaps suggest also that they sort the sale of their own house out, as that has been adding considerably to the stress. (I personally would also be tempted to make the point, tactfully, that she will then be able to make sure that everything is carried out exactly as she wants it, and this will avoid the possibility of any further unfortunate misunderstandings. My best guess is that the prospect of that extra work will make her reconsider some of her previous words!)
At the end of the day, it is a difficult situation. Do what you feel to be right. For you, and your immediate family. Longer term, a bit of diplomacy, may make life further down the line a bit easier. But you may feel that isn't true to you, or congruent with your deeper values. Treasure what's important to you, try and let go of what isn't.