Kemp made his announcement in the governor's office of the Georgia Capitol, standing beside the man he plans to replace in January. Republican Gov. Nathan Deal called Kemp "the governor-elect" and both said they would begin transition work together.
"We won a clear and convincing victory," Kemp said of returns showing him with 50.3 percent of almost 4 million votes, about a 63,000-vote lead over Democrat Stacey Abrams. That's a narrow sum considering the near-presidential election year turnout, though sufficient for the majority required for outright victory.
Abrams maintained there are enough uncounted ballots to force a December runoff in one of the marquee matchups of the 2018 midterm elections.
The Associated Press has not called the governor's race.
With legal wrangles opening on what votes to count and how, the dispute is prolonging a bitter contest awash in historical significance and national political impact. Abrams hopes to become the first black woman elected governor of any American state. Kemp seeks to maintain Republican dominance in a growing, diversifying Deep South state positioned to become a presidential battleground.
The key question is how many uncounted ballots actually remain. Kemp says it's less than 21,000 - almost certainly not enough to force a runoff. The elections chief from the secretary of state's office said in a federal court hearing Thursday afternoon that the number is 21,190.
"Even if she got 100 percent of those votes, we still win," Kemp told reporters.
Abrams' campaign argues the total could be higher, and the secretary of state's office has been scant in sharing details as officials in Georgia's 159 counties keep counting.
"This is about the integrity of the election in the state of Georgia," said Abrams' campaign manager Lauren Groh-Wargo. "Brian Kemp can't just walk away from that ... Our governor (Deal) can't just walk away from that."
John Chandler, one of several attorneys for Abrams, promised to "litigate until we have determined that every person's vote has been counted."
A runoff, if needed, would be Dec. 4.
County authorities must certify final returns by Tuesday. The state must certify a statewide result by Nov. 20. Deal appointed one of his Cabinet members to oversee the process in Kemp's place.
Abrams, other Democrats - including former President Jimmy Carter - and voting rights activists had for months called for Kemp to step down amid charges he was abusing his office to make it harder for some Georgians, particularly minorities, to vote.
Kemp said his resignation "will give confidence to the certification process." He maintained he wasn't bowing to pressure but preparing to be governor.
"That was all political," Kemp said of previous criticisms, adding Tuesday's turnout - about 1.4 million more than in Deal's last election - proves it.
One of the lawsuits heard Thursday in federal court requested Kemp be barred from overseeing the rest of the certification procedure - a requested pre-empted by Kemp's resignation.
Abrams' campaign said it believes she needs to pick up about 25,000 votes to force a runoff.
Offering examples of potential ballots Democrats say Kemp isn't contemplating, Groh-Wargo said four counties reported considerably fewer early votes in the governor's race than the number of early ballots cast. Groh-Wargo said it seemed implausible that voters cared enough to cast ballots early but not in the hotly contested governor's race.
She added officials in suburban Atlanta's Cobb County added several hundred votes to that count Thursday morning from absentee ballots. That came after the secretary of state's office said all absentee and early ballots had been counted.
Abrams' lawyers also said they planned to sue officials in Dougherty County because absentee ballots were delayed after Hurricane Michael devastated parts of south Georgia. Separately, the ACLU raised concerns over 1,200 absentee ballots in Gwinnett County, northeast of Atlanta, which it said were rejected because of missing birthdate information.
"Brian Kemp owes voters an explanation," Groh-Wargo said, demanding to see lists and names. "We do not believe any of these numbers are credible."
When Kemp's campaign declared victory Wednesday, aides cited a statewide estimate of uncounted ballots from the secretary of state's office. But that office had not publicly offered a county-by-county breakdown to Abrams' campaign or the media at that point.
The provisional vote total is considerably higher than in 2016, when a slightly larger electorate yielded 16,739 provisional ballots. Of those, 7,592 were counted.
State and campaign officials have said they expected a much higher proportion to be counted this year. In federal court Thursday, a secretary of state's representative said the provisional count included at least some votes cast late at a handful of metro Atlanta precincts that courts ordered to stay open past Tuesday's 7 p.m. poll closing time.
Kemp said he and Abrams have not communicated since the election. And while Kemp said he respected Abrams' efforts, he declared Thursday, "The votes simply are not there for her."
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