Donald Trump’s campaign declined to comment Thursday on whether North Carolina Lt. Gov. Mark Robinson, whom the former president once described as “Martin Luther King on steroids,” should drop out of the battleground state’s race for governor.

North Carolina Lt. Gov. Mark Robinson, the state’s Republican candidate for governor, said he would stay in the race and denied new accusations that he posted racist, antigay and inflammatory comments on a pornography-website message board more than a decade ago.

Mr. Robinson, the Republican nominee for governor of North Carolina, vowed to stay in the race and sought to deny the report, which also said he had defended slavery in online posts years ago.