Legally, an announcement by one adult that s/he wants to "sever ties with" another has effectively no meaning.
Presumably in the case mentioned the daughter intends to stop interacting with Musk, or significantly limit such interactions. She is free to do so. That has no legal effect on Musk's will. It might cause Musk to change his will, but that is his decision. He is free to leave money, even lots of money, to a person who has "severed ties" with him if he so chooses. So is anyone else. (Of course most people don't have as much to leave as Musk presumably will.)
If Musk does choose to leave money to this daughter, her announcement of negative feelings towards Musk would not be grounds to challenge the will or to disinherit the daughter. There are various grounds on which a will may be challenged in US law. Perhaps the most common are undue influence, and lack of the needed mental competence. Announcing an intent to "sever ties" would not be a usual way to obtain undue influence over a person, and nothing in Musk's reported actions shows the kind of lack of mental competence which would render a will void.
In short, the only way in which this or any similar announcement might change Musk's will is if it first changes his mind.
As for the daughter challenging the will, she would have no more and no less rights to do so than if no announcement had been made. There are various legal grounds on which a will may be challenged, depending on the state. They are all fact-dependent, and can be done only if certain sorts of facts are alleged in the challenge. There is no way to know, now, if any such facts will be true when Musk dies. But the announcemt on its own will not matter.