The allegations, first reported in October, said he tried to obtain personal information about his accuser Virginia Giuffre through his police protection.
The health secretary should focus on offering a deal instead of "scaremongering the public", the BMA says.
In a TV broadcast on Friday night, the King said an early diagnosis was key to the "good news" that his treatment is being scaled back.
Fire crews called to reports of an explosion on Friday evening.
Broadcaster Emma Willis will take over presenting the Saturday afternoon programme.