The First Minister claimed millions of pounds could be lost in funding for research trials after leaving the EU.
She pointed to other dangers, such as pharmaceutical companies being told to stockpile medicine in preparation for a no-deal scenario.
Speaking ahead of her speech at the National Cancer Research Institute Conference, the First Minister said: “Far from the £350 million extra a week promised to the NHS from Brexit, there is growing evidence of mounting costs.
“It’s now time the UK Government came clean on the scale of the cost to our health service.
“The Prime Minister failed to guarantee there would be no interruption in medicine supplies after Brexit.
“This came after UK ministers published a truncated tender, at a cost of tens of millions of pounds, associated with medicines stockpiling.
“There is no good Brexit for our health service. Even with a deal we face a loss of millions of pounds in research funding which could hit clinical trials.
“The best way to avoid this damage to the health service is to remain in the EU.
“But short of that, the UK must stay in the Customs Union and Single Market - which is around eight times the size of the UK market alone - to minimise the damage of Brexit and ensure we continue to have access to the high quality frontline and research staff we need.”
She added that the Scottish Government is doing all it can to mitigate the expected damage Brexit could cause the NHS.
A spokeswoman for the UK Government’s Department of Health and Social Care said: “The Government is confident of reaching a deal with the EU that benefits patients and the NHS - but we are preparing for all situations and we are working closely with partners to ensure the proper provision of all medicines in the event of a no deal Brexit.
“The department is working closely with pharmaceutical companies to minimise costs and we have received good engagement from the industry so far.”