Many showing their respects to the Edinburgh-based firefighter have taken to social media to post an image of a black circle with a red line through it - a symbol used by fire departments to show respect for firefighters injured and killed in the line of duty.
Mr Martin, 38, from Fife, sustained serious injuries during the blaze at the former department store which started on Monday.
He was one of five firefighters taken to hospital during the fire.
He died on Friday, making him the first Scottish firefighter to die in the line of duty since Ewan Williamson, who died after being injured in Edinburgh in 2009.
Fire departments across the UK have paid their respects, including Ixworth Fire Station in Suffolk, who said: "We would like to send our deepest condolences to the family and friends of Firefighter Barry Martin who has sadly lost his life after being injured whilst fighting a fire in Edinburgh."
Members of North Wales Fire and Rescue Service said their thoughts were with Mr Martin's family and the Scottish firefighting community.
While South Yorkshire Fire and Rescue paid tribute saying: "Our thoughts and deepest sympathies are with family and friends of firefighter Barry Martin and everyone at the Scottish Fire and Rescue Service.
"We are so sorry for your loss."
First Minister Nicola Sturgeon tweeted her condolences, adding that the tragedy was a "reminder of the selfless courage our firefighters demonstrate in the line of duty each and every day".
Health Secretary Humza Yousaf paid his respects to Mr Martin and his family, and thanked all emergency services "who often confront danger to keep us safe."
Leader of the Scottish Liberal Democrats Alex Cole-Hamilton called for Mr Martin to be posthumously awarded the George Cross in the wake of his death.
The MSP said the 38-year-old's actions "went above and beyond the call of duty" before adding: "I will be making representations for Barry to be awarded the George Cross posthumously in honour of his extraordinary bravery."
City of Edinburgh Council's leader Cammy Day also paid tribute to the firefighter, saying: "I'm incredibly sad to hear about the death of firefighter Barry Martin.
"My heartfelt sympathy goes out to his family, friends and colleagues. It's at these tragic times that we're reminded of the immense bravery and selflessness of our firefighters and emergency service workers - putting their lives on the line day in, day out to keep us safe.
"I know this will be an extremely difficult time for many people and my thoughts are very much with them all."
Edinburgh shop Hanover Healthfoods announced it was making a donation to the Firefighters Charity in the wake of Mr Martin's death "so that all firefighters can feel our appreciation for the risks that they take to keep us all safe".
ScotRail also tweeted: "We are very sad to hear of Firefighter Barry Martin's passing.
"On behalf of all at ScotRail, we offer our sincerest condolences to his family, friends and colleagues."
Ross Haggart, the Scottish Fire and Rescue Service's interim chief officer, said he and his colleagues were "devastated" at Mr Martin's death.
He said Mr Martin's family and the service have been "overwhelmed with the messages of support we have received and we thank everyone for the time they have taken to share these."
Founded in 1838, the department store was one of the oldest in the world when it closed in 2021.
The A-listed current building dates to 1895, after a fire destroyed the original.
Renovations are ongoing to transform it into a hotel, backed by a firm owned by fashion billionaire Anders Holch Povlsen.
A total of 22 fire appliances were sent to the scene at Rose Street after the fire was reported at about 11.30am on Monday.
Six emergency workers were taken to hospital because of the fire, which included five firefighters and one police officer.
Two firefighters were treated for burns, and a police officer and two firefighters for smoke inhalation. They have since been discharged.
Superintendent David Robertson of Police Scotland said: "Our thoughts are with the family, friends and colleagues of Barry at this very difficult time.
"Our inquiries are ongoing to establish the full circumstances."