What are my rights as it pertains to withdrawing a resignation?

My fiancée, a 30 year employee of his company, sent an email to his supervisor indicating that he will likely resign due to the pressure of the newly instituted regulations and requirements from a relatively new owner but made it clear that he would stay until a replacement was found and trained. So, no definitive timeframe was offered. 4 days later, he spoke with the owner of the company, who was on vacation during the time that he sent the email and had just returned. They discussed the issues and it was agreed that he would stay on. Nothing was offered verbally or in writing to the effect that his resignation was received or accepted, however, it has been a month since this conversation took place and recently, the owner asked him how things were going. He said he was working hard at doing everything that was asked of him and would do so as long as he has a job to do there. The owner responded, "Who said I never accepted your resignation?" Can he legally leave it hanging like that?