The county's elections board voted in February to fire Barron, following a remarkably contentious election season.
Fulton's Board of Commissioners, however, did not endorse the move, leaving Barron in the position.
On Thursday, with Fulton again in the spotlight after Tucker Carlson focused on alleged election irregularities in the county in a report on his Fox News program on Wednesday night, Raffensperger made the call for Barron and Jones to be fired.
"Fulton County's continued failures have gone on long enough with no accountability. Rick Barron and Ralph Jones, Fulton's registration chief, must be fired and removed from Fulton's elections leadership immediately," Raffensperger tweeted. "Fulton's voters and the people of Georgia deserve better."
Raffensperger is facing a Republican primary challenge next year from supporters of former President Donald Trump, including Rep. Jody Hice, who endorse the president's unfounded claims of a concerted effort in November to fraudulently hand Georgia's election to President Joe Biden.
In a tweet, state Rep. Bee Nguyen - who has declared her candidacy in the Democratic primary next year for secretary of state - countered Raffensperger, writing: "By his own admission, we conducted the most secure election in Georgia’s history - but here he is again trying to appease a radicalized & dangerous base that has rejected him. We deserve better."
Carlson's program on Wednesday night focused on a report in the Atlanta Journal-Constitution of a limited number of ballots that were scanned twice before Georgia's machine recount in November - which followed the initial machine count immediately following Election Day and a hand-counted audit - and on discrepancies in the audit.
"There’s no indication any vote for president was counted more than once in official results," the AJC's Mark Niesse reported.
In the hand audit, digital ballot images from Fulton County reveal discrepancies with some results in the secretary's official audit report. Raffensperger maintained at the time of the hand audit that human error would produce discrepancies with the official machine-count results, and that its purpose was to provide a general indication that the machine results did not produce a glaring error.
Those hand audit results did not fully match machine results in many counties throughout the state, but did not differ in large degrees.
As the Fulton County elections director, Barron became one of the most visible faces throughout Georgia's post-election saga in November and December. The county faced criticism over issues regarding election security, staffing and resources and ballot chain of custody.
Sec. Raffensperger has frequently characterized Fulton County as a problem district in terms of election administration, even as he sought to stress the overall legitimacy of Georgia's election results.
Barron, in turn, has defended his performance by saying the county has received insufficient support from the state.