A recount is continuing in the race between Mr Nelson and his challenger for the Senate seat, outgoing Republican Governor Rick Scott.
Gillum, in a video that he posted on Facebook, congratulated Republican Ron DeSantis but vowed to remain politically active although he gave no clues as his future plans.
"This has been the journey of our lives", said Gillum, appearing in the video with his wife, R. Jai Gillum.
"This was about creating the kind of change in this state that really allows for the voices of everyday people to show up again in our government and our state and our communities", he said.
Gillum conceded on Election Night, but retracted it after the margin of votes between him and DeSantis shrunk within the level of a legally required recount. But after that margin further narrowed in the days after the midterms to within a half of a percentage point, and a statewide machine recount was triggered, Gillum held a brief press conference last weekend in which he withdrew his concession and called for counties to "count every vote". A second, manual recount is underway in that contest, because the margin following the initial machine recount was still too narrow by state standards.More news: 'Mad' Harry Redknapp joins I'm a Celebrity
The concession ended Gillum's first bid for statewide office in Florida, but the run - despite his loss - vaulted the 39-year old politician into the upper echelons of Democratic politics.
At least 44 out of 67 counties have finished their hand recount in the Senate race. "We need to update Florida's election system and bring it into the 21st century", Gillum said.
For the Senate race, unofficial results show incumbent Nelson trailing Scott by about 12,600 votes out of more than 8.2 million cast, or 0.15 percentage points.
The South Florida Sun-Sentinel posted video Saturday of Broward County Supervisor of Elections Brenda Snipes saying that 2,040 ballots had been "misfiled".
Officials have until noon on Sunday (local time) to tally any votes missed by electronic voting machines.
The drama of counties across the state recounting ballots brought back memories of Florida's 2000 presidential recount. Snipes did insist that the ballots were still in the elections building.