@Noir4241 or spoken more specifically to the instance, she called the prince on his actions—he was asked to act as interference against the Prime Minister's daughter (or whatever the Hell that Duke is), thereby seeking to remove the "hillbilly" by embarrassing her and making her leave to clean her dress.
Instead, she called the prince on his actions (by spilling her drink onto him in return, she called his bluff, so to speak). The Duke realized that his daughter stood no chance against such a fearsome spector as the 31st consort, and that if held in open combat, his daughter would likely would be her own ruin if not risking worse for his house. Best she withdraw as the longer she stays to cause problems, the more likely her rank at court would be effected as well as wasting time given her age (older in this case = less desirable as a prospective marriage partner for a woman of her class) in the hunt to sell her off to the most bountiful prospect (most desirable bachelor). This precludes that the Duke has some other source of heir and that in this society, women cannot inherit a title outright. Amongst other factors.