The incident prompted Secret Service Director Mark Sullivan to issue a new code of conduct that included barring employees from drinking within 10 hours of the start of a shift.
It is unclear when Engler was scheduled to work again. He did not immediately respond to a telephone message seeking comment.
In August, a Secret Service agent left a gun in the bathroom of Republican presidential nominee Mitt Romney's campaign plane and a reporter found the weapon. The Secret Service said the agent was assigned to Romney's security detail.
Miami police say Engler was arrested on two misdemeanor charges and released to members of the Secret Service's Miami field office. It's unclear where he is based.