State law requires a machine recount in races where the margin is 0.5 percentage points or less.
"Nobody wanted to be governor more than me, but this was not just about an election cycle", Gillum said in the video posted Saturday.
The state's similarly close U.S. Senate race between Democratic incumbent Bill Nelson and Republican Gov. Rick Scott remains officially undecided.
Democrat Andrew Gillum, who had sought to become Florida's first African-American governor, has conceded after a recount showed he had no way of catching his Republican rival Ron DeSantis, an ally of United States President Donald Trump.
'Congratulations to Andrew Gillum on having run a really tough and competitive race for Governor of the Great State of Florida, ' Trump tweeted Saturday.
Gillum, whose refrain had been "bring it home" as he recounted stories of growing up poor in the state, concluded his campaign with a Facebook video he recorded alongside his wife in a park.
After Gillum initially conceded the contest on election night, his subsequent calls for every vote to be counted echoed similar appeals from fellow Democrat U.S. Senator Bill Nelson.
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"We said that we would fight until the last vote is counted".
"Campaigns are meant to be vigorously debated contests of ideas and competing visions for the future", DeSantis said. "This fight for Florida continues". I believe those ballots were probably mixed in with another stack, ' Snipes told the elections Canvassing Board at midday.
The South Florida Sun-Sentinel posted video Saturday of Broward County Supervisor of Elections Brenda Snipes saying that 2,040 ballots had been "misfiled".
State officials ordered a manual recount earlier in the week after a legally required machine recount showed that Scott led Nelson by about 12,600 votes.
At least 44 out of 67 counties have finished their hand recount in the Senate race.
Officials have until noon on Sunday (local time) to tally any votes missed by electronic voting machines.
It was another snag for Snipes, who has already been under fire for the way her office has handled the election and recount.