The article appears on the website of the Sinn Fein youth organisation, Ogra Shinn Fein, and shows a photograph altered to depict the head of the DUP man super-imposed on the body of Joseph Goebbels - one of history's most reviled figures.
Mr Campbell, the North's Minister for Culture, described the comparison between himself and Goebbels as "grotesque".
"I think if a serious political party wishes to make an objective criticism of opponents, then this is not the way to go about it," he said.
"To compare a democratic opponent to a Nazi, particularly a chief propagator like Goebbels, is wrong particularly when the republican movement has been steeped in fascism and would be considered by many to be not just grossly offensive but on the border of being totally unacceptable."
Sinn Fein MLA Barry McElduff said his party would be discussing the issue later today (Wednesday).
"If he has sensitivities, if he has taken issue with something, out of respect I will convey that to other Sinn Féin people when I meet them today," he said.
"Gregory has shown a lack of respect, and even intolerance, towards the Irish language and Irish culture, which is regrettable on the part of the Minister of Culture."
Mr. McElduff said the issue also raised concerns over the use of people's images on the internet.