The Evening Times was praised by British Heart Foundation Scotland, the Scottish Kidney Federation and the Cystic Fibrosis Trust Scotland after yesterday's event.
The British Medical Association (BMA) said the time was now right for the Scottish Government to lead the public debate and believed the Scottish public will support the change.
When the last poll was carried out in 2008, around 72% of the UK public said they were supportive of an opt-out system. The BMA believes the figure could now be as high as 90%.
Dr Sue Robertson, a renal doctor and BMA spokes-woman, said: "The time is now right for the government to lead the public debate. The work that has been done has helped the situation but there is still work to be done and opt-out is the next step.
"I think the public will be supportive in Scotland if it is explained to them properly and, if the safeguards are prepared, the public will support it."
Dr Robertson said work needed to be done to allay fears about organ donation under an opt-out system.
She said: "It is a real fear. People are frightened that there might come a time when doctors see a dying patient as a source of organs, so we have to allay this fear.
"Introducing a soft opt-out system makes it easier for relatives to make that decision and allows us to respect the wishes of a dying patient.
"Gradually, over time, society will start to see the benefit and organ donation will become the norm."
A spokesman for British Heart Foundation Scotland said: "BHF Scotland congratulates the Evening Times for its very successful campaign.
"Clearly, there is a huge amount of public support for a 'soft' opt out system of organ donation and we urge the Scottish Government to introduce such a system without delay."
Alison Blezard, of the Scottish Kidney Federation, said: "This is a very important next step in the campaign for an opt-out system."
Yvonne Hughes of the Cystic Fibrosis Trust (Scotland) praised the Evening Times for progress made so far. She said: "The Evening Times has opened up the public debate. We are in favour of anything that makes more organs available for transplant, including a soft opt-out system."
HOW TO HELP
Sign our petition to lend your support to an 'opt-out' system for organ donation at www.eveningtimes.co.uk
You can also sign up on Twitter and Facebook via http://bit.ly/etoptin
To join the existing national organ donor register, text 'ETDONATE' to 61611.
Texts are charged at the usual standard network rate (any texts sent in reply are free of charge to the recipient).