I have to be honest and say that I can't help but empathise with your sister; she clearly loves you very much if she got that upset about not getting to spend time with you that she sent such an angry message to her friend. Of course she shouldn't have called you that, and I very much doubt that she meant it, but it shows how much you, and getting to spend time with you, mean to her.
She's probably embarrassed and a bit humiliated that you saw the text (hence hanging up), and can't turn around and say "I love you so much and just miss you and feel a bit sad/lonely/jealous when I'm confronted that you have other priorities" because, as siblings, we often don't speak that openly, instead resorting to childish namecalling and passive aggressiveness. It's also possible that she's not emotionally aware enough to know why the pain of you not having time for her triggered her to speak badly about you.
I think the mature way forward would be for you to give the olive branch and say "I really value our time together just as much as you do, and love you very much, but I have also got to put one-on-one time into my marriage, which in no way detracts from the fact that I'm always going to be there for you, and put time into you and your DC, no matter what."
I know there's the whole "don't reward bad behaviour" line, but to be honest, life is too short, especially when it comes to family.