Had she won, Abrams would have become the first black woman elected to be a state governor in US history.
The speech Abrams delivered at her campaign headquarters Friday evening marked the close of the 44-year-old attorney and former lawmaker's unsuccessful attempt to make history as America's first black woman governor.
Abrams also announced she will launch a new voting rights group called "Fair Fight Georgia".
Abrams has been embroiled in a legal battle with Kemp and the state of Georgia ever since the midterm elections on November 6.
"I will not concede because the erosion of democracy is not right", she said.
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Unofficial returns in the state put Kemp ahead of Abrams with roughly 50.2 percent of more than 3.9 million votes. "But to watch an elected official who claims to represent the people in this state baldly pin his hopes for election on suppression of the people's democratic right to vote has been truly appalling".
"Let's be clear, this is not a speech of concession", she said.
As secretary of state, Kemp also served as Georgia's chief election officer and refused to resign or recuse himself from his position until two days after the general elections.
Stacey Abrams' campaign is preparing an unprecedented legal challenge in the unresolved Georgia governor's race that could leave the state's Supreme Court deciding whether to force another round of voting.
Ms Groh-Wargo said on Tuesday that the Abrams campaign believes she needs a net gain of 17,759 votes to pull Mr Kemp below a majority threshold and force a December 4 run-off.
"I don't want to hold public office if I have to scheme my way into the post because the title of governor isn't almost as important as our shared title - voters", she said.
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Kemp, who until last Thursday was Georgia's secretary of state and in charge of overseeing its elections, declared himself the victor on November 6, but Abrams has yet to concede the race.
It was, however, the end of a campaign whose outcome had remained uncertain for days as Abrams pressed for the counting of ballots that had been rejected for minor errors.
Abrams' campaign sparked huge energy across the state and she became a national Democratic star.
The razor-thin margin between Abrams and Kemp is reflective of Georgia's march from once-solid Republican terrain toward becoming a purple state. Ironically, this time around, it seems Democrats are having a tough time coming to terms with election results.
Affidavits from poll workers reviewed by the AP describe long lines that discouraged people from voting, poll workers failing to offer provisional ballots to people who didn't show up on the rolls or were at the wrong polling place and election equipment that froze and had to be rebooted.
The election has been mired in accusations of election fraud and voter disenfranchisement. The final result had been in doubt for 10 days after the election. Trump won Georgia by five percentage points in 2016, but Democrats' increasing strength in the suburbs bodes well for their chances in future statewide races.
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