My wife has been working at the same UK office for about five years but recently the atmosphere has turned toxic. Some of it is due to increased workload for all involved, but most of it is caused by a new manager's inexperience.
Due to work problems my wife has seen a doctor, who recommended her to take time off work. She has no problem with that, but since she is planning to leave the place it worries her that this would be added to her sick record and may become an issue when applying for new jobs.
My assumption is that the answer is "no, they can't", unless you tell the prospective employer yourself. But since it is just an assumption I wanted to ask people who are likely to be better informed than I am..