What you have done.
You have done damage to the person possibly quitting, and you have done utter damage to your own reputation. You have done damage to the company, too, considering the wasted work time, and probably there is more damage to come to the company.
More of a sidenote: If the company is ethical and interested in a good team spirit, you might just have put yourself on the ramp. Where there is a good spirit, there is not place for unfounded allegiations.
Remarks:
It was not your business. It's his/her and the company's business to negotiate an appropriate cancellation period. And as long as both sides honor the laws in that regard, it's neither anything YOU can do or influence, nor is it in any regard your business.
You do not even have proofs. From a coworker's point of view, you may make a a joke of yourself. It could be that the other guy has cancer or is burnt out, and in the utter need of a therapy. He/she might even be donating his leave days to not cause friction with the company targets. Or maybe his/her significant other currently is in the need of help on certain days; a newborn might be approaching, or significant other may need a drive to the distance university because of exams or tests.
What if the company starts interviews and even hires someone new, and it turns out he/she was simply going preparing for an upcoming Karate event? Now you have done damage to the company and to the newly hired guy who must now been told that all her/his other interview cancellations were waste! ** Or, the company decides that this is a crappy situation, and fires your coworker based on your "observations", and now his approaching newborn cannot even be fed or his/her significant other cannot get any more treatment against cancer? Or misses exams or must quit university?
What is your benefit, really?
In no case, I can see any benefit from what you've done. You have not the slightest clue of what could be, and you have not the slightest clue of what can happen because of your allegations.
Expect to become an avoided person in the future and not be told anything sensitive again. Alternatively, I hate to say, you are now in the danger of becoming a target of bullying.
Footnotes:
** All mentioned examples are things I would not necessarily tell any random coworker.
What you can do.
- Tell the coworker what you have done and sincerely apologize. He will knock you out for this, but if you are a real man/woman, you will do this.
- Apologize to everyone you have told, and explain that any guessing from your side was asocial and pure speculation, and not well thought.
- Tell them you erred for causing any difficulties on this and promise to avoid making baseless allegations in the future.
However, the bobsled has left the ramp, and you probably cannot improve the situation anymore. Let this be a lesson for life.
Finally. Answering your question.
Can/Should I do anything about Joe possibly leaving?
If you could have done anything in the past, now you cannot anymore.
If he leaves, don't get into the idea of following him; he will not want to have you as a coworker ever again if he knows (or find out later) what you did.
