His resignation was prompted by his strong disapproval of the planned five-day strike by resident doctors in England, set to commence on 25 July.
Mr Winston, an 84-year-old Labour peer, said that the "highly dangerous" walkout, triggered by a dispute over pay, could erode public trust in the medical profession.
He said that strike action disregards patient vulnerability and urged the BMA to reconsider, warning of potential "long-term damage" to faith in doctors.
Health Secretary Wes Streeting also appealed to resident doctors to abandon their "unreasonable rush to strike", cautioning that the NHS recovery remains "fragile".