The committee's report said: "We do not rule out introducing presumed consent in Wales at some point in the future.
"However, we do not believe that it is currently the most urgent priority and believe that it could be a distraction from other more productive actions."
Committee chair Jonathan Morgan said: "Asking grieving people for their consent to remove part of their loved ones, even if it is to save the life of another, is not an easy task.
"So I can see the attraction of presumed consent. Unfortunately, whatever its attractions, it's not a silver bullet."
The BMA said the decision was a 'lost opportunity' for change
The report recommended that any presumed consent system in the UK should be the "soft" version, where loved ones were consulted and their views taken into account.
It also called for review of Wales' transplant capacity and to increase the number of intensive or critical care facilities in Wales to help increase the number of organs available for transplant.
BMA Wales Cymru welcomed the calls to boost transplant capacity but disagreed that going for presumed consent would act as a distraction.
The BMA's Welsh secretary Dr Richard Lewis said: "We feel the committee has missed a real opportunity for Wales to take the lead and send a clear message to the rest of the UK that a 'soft' system of presumed consent would increase donation rates and save Welsh lives."
The three committee members who voted for a presumed consent system were Plaid Cymru's Dr Dai Lloyd and Helen Mary Jones, and Lib Dem Jenny Randerson.
Dr Lloyd said: "Quite simply, people are dying on these waiting lists. We have the opportunity to change that by simply changing our approach.
"There is no doubt that a system of presumed consent would produce a far higher potential donor rate than at present, which is far too low at 22%."
About 470 people in Wales are waiting for a kidney transplant and around 780,000 people are currently on the organ donor register in Wales.
This week, the family of Meg Burgess, three, who died after a garden wall collapsed onto her, described donating her eyes and heart for transplant as "the greatest gifts anyone could ever receive".
A survey by the committee found nearly half of those questioned said they would like their organs taken automatically after their death to save a life.
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