Dear Nishu, please note that the submission of resignation is an official matter, nothing personal about it. Therefore, if her colleagues come to know about it and start asking questions, the sky did not fall on her. Anyway, HR would have started searching for her replacement, and the bubble of this secrecy would have burst in a few days or weeks.
Think of the opposite for a while
When a newcomer joins an organization, other colleagues go to his/her desk and give a welcome. Should a newcomer complain about it? Generally, when a person starts looking for a job, it is known in close circles. In many cases, even managers know about it. Probably, the woman employee's resignation caught them unaware. Employees always form opinions about others. By one's activities or way of working, they know who could continue and who could quit. Possibly, the woman employee was considered as one who has strong allegiance to the organization. The news of her resignation punctured their belief, hence the expression of surprise! However, this astonishment would have fizzled out anyway in a day or two. Yes, for a resigning employee, it is irksome to explain the reason to every person, but then she could have shown some patience. Rather than prying about her resignation, similar patience her colleagues also could have shown.
You have written that the resigning employee's manager and HR knew about her resignation
But for the leakage of the news, why are you holding HR responsible? What is the guarantee that her manager did not divulge the news?
For Rahul Chhabra
You have recommended action against HR. You have written that "It is highly immoral of that HR person who has disclosed news of her exit." What is immoral about it? Has HR disclosed anyone's illicit love affair? Immoral means not adhering to ethical or moral principles. Yes, HR is expected to keep the affairs of the organization to themselves, but by chance, if they have divulged it, even then, this disclosure cannot be called "immoral."
Going further
You have written that "If this disclosure has an impact on the employee or her dignity, then a formal complaint should be made to the management." How can the disclosure of news of one's resignation pull down one's respect or esteem is not understood. As far as a complaint to the top management is concerned, they could consider it as churlish. They have a lot of tensions about business than worrying about anyone's resignation. Nothing wrong with expressing one's opinions in the public forum or taking a position, but then one should be mindful of one's words. Please remember that the use of wrong words could lead to misguidance also!
Thanks,
Dinesh Divekar