I’m annoyed that this gossiping has happened
It might have been gossiping, but it doesn't seem to have been.
I questioned where they had heard this and they say our manager has told them.
This seems normal, especially in the context of e.g. a team meeting.
You have no right to confidentiality about your resignation. It is an official company issue, and representatives of the company may inform whoever they wish, for any reason. In some cases they will be required to do so, such as tax office. In other cases, it may be necessary for planning.
Usually there is some etiquette around resignations, which varies by region and company internal culture. In my experience, often managers will respect a request that you inform colleagues first, and will either allow you to announce it yourself or share it themselves equally in some group meeting. In some cases, managers prefer to keep a resignation quiet until a "good time".
It is possible your manager has breached etiquette for your company or region. But they haven't breached any right of confidentiality purely by mentioning that you were leaving. Even if you explicitly state that your resignation should be kept confidential, this has no real weight except that the manager might agree to your request.
If there is extra information around the reason for you leaving, that might be protected. And it would be poor etiquette to frame it as you being fired if you had not been for example (in some cases this could be slander, or libel when written down assuming it is not true). You don't go into detail in the question, but that does not seem to be the case here.