Clyburn said during a news conference that a vote for fellow Democrat Alvin Greene would be an insult to his three daughters and granddaughter.
Greene is accused of showing pornographic images to a female University of South Carolina student.
"I always ask people how would you feel if it were your child," Clyburn said.
Greene, an unemployed military veteran and political unknown, easily beat a party-backed former state lawmaker and judge in the June primary.
"Everything about that candidacy is illogical," Clyburn said.
State party officials have asked Greene to step aside because of the indictment.
"According to state law if you're indicted and you hold public office, you are immediately removed from office and I think the same thing ought to apply to anybody in pursuit of office," Clyburn said.
Phones messages left for Greene were not returned, but he has said in the past he plans to stay in the race.
Greene faces Republican incumbent Jim DeMint and Green Party candidate Tom Clements in the general election in November.
Clyburn plans to vote for write-in candidate Mazie Ferguson, a longtime Democratic Party activist from Sumter.
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