Ms Sturgeon told MSPs last year that the Scottish Government had no plans plans to change to opting-out, however it had not ruled out legislative change in the future.
Labour's health spokeswoman, Jackie Baillie, said: "Nicola Sturgeon failed to agree to change the legislation which is widely thought would save many more lives and bring hope to the thousands of Scots who are currently waiting for organs on the transplant waiting list.
"Alex Neil has previously stated his support for such a change which has long been supported by patients groups and the medical community. We hope that he will now use his new post to make it happen.
"We were disappointed that there was nothing in the legislative agenda announced last week to bring this proposal forward. We will support the government in introducing legislation which could be life saving for so many Scots and I hope Alex Neil will now bring forward proposals as soon as possible."
Dr Brian Keighley, chairman of the BMA in Scotland, said: "We welcome Labour's support for an opt-out system for organ donation to be introduced in Scotland. New ways must be found to increase the number of organs available to those who desperately need them and an opt-out system, with safeguards, would better allow that to happen."