CARE spoke out as a debate was secured in Parliament on Thursday by Labour MP Dan Jarvis about taking a presumed consent approach.
CARE's Director of Parliamentary Affairs, Dr Dan Boucher said: "CARE does not believe that the state should have an automatic right to a person's individual organs.
"This approach undermines organ donation as a genuine gift and goes against the consensus that human beings have autonomy over their own bodies."
"It also risks provoking a significant minority of people to withdraw themselves from donation, thereby reducing the stock of potential donors. In a context where people are dying every week for lack of available organs we cannot afford to do this."
People living in England currently have to opt-in for organ donation. Under a presumed content scheme, families of people who opt-out would not be consulted over their loved ones' organs.
The organisation says initial figures from Wales, where an opt-out system was introduced in December 2015, are "not encouraging".
They show that rather than leading to an additional 45 organ donations each year as hoped, just 33 organs were donated between April 2016 and April 2017.
Statistics also show more than 170,000 people withdrew themselves from the donation register in Wales. In the same period, the figures for England, Scotland and Northern Ireland were 27,559, 1,834 and 204 respectively.
CARE is calling on the UK to follow the lead of Spain - which has the highest organ donation rates in the world - by spending money on educating people about the merits of organ donation.
Dr Boucher added: "CARE wholeheartedly supports the reasoning behind wanting to secure this debate; the numbers of people waiting for transplants is painfully high and more must be done to increase the pool of potential donors"
"Based on organ donation data from the first full year of the opt-in system in Wales, the message for other UK jurisdictions tempted by the an opt-out system is - for the moment at least - leave well alone."