This may be one of the few rare times where a prince annulling his engagement is justified, but seriously, do they always have to do it in front of everyone?
Public humiliation.
Terrible thing, since it practically label the victim as "okay to bully". They want their enemy to be stripped and destroyed, to make the trauma as deep as possible.
There is a logic, tho lost to time by now, to it. Essentially, the princes don't actually have the power to annul their engagements, those were set up by their parents, the sovereigns. But make a public commotion of it with rumours abounds, it would undermine the royal authority and cause an even bigger scandal, if the monarch then revoked the revoking.
Rumours (which nobody knows the original source) is still okay, but this kind public stunt out of nowhere is too risky for him.
Rosemary might be commoner born, but she's still legally the daughter of a duke's family.
Nobles don't like losing face, and if we disregard the Violets not having much allies in this story, picking a fight against a high ranked nobility is picking a fight against their whole political faction. Einrich being from the royal family (means not one of the nobles) made him need to watch his steps even more carefully.
Also, even if he fulfilled his objective to cancel the engagement (and pressure his father to approve it because of the public opinion), the nobles already see a member of the royal family breaking a deal unilaterally without any room to talk. That would hurt their trust and loyalty to the royal family. Like, every one of them must have a dirt on themselves. Who to say the royal family wouldn't suddenly bring it up as pretext to play the righteous one.
The wise move would have been to already gather support from many nobles before executing, so he will have people backing him. Or better, discuss it with his father first so the adults can make the necessary arrangements. Dealings are done behind closed doors, the signing is just public display.