Tributes pour in for the iconic leader's 90th birthday.
The decision by network bosses came as the Obama campaign launched a website to dispel rumours about his faith and patriotism and his wife's views on race as he prepares to compete for the White House in November.
The website, www.fightthesmears.com , offers detailed responses to several rumours that have continued to circulate online and in conservative news outlets, including that Mr Obama is a Muslim.
E. D. Hill, a veteran of the Fox network, made her comment after the presumptive Democratic nominee and his wife affectionately bumped fists on stage last week as he prepared to make his victory speech.
Before a commercial break on her America's Pulse show, she asked: "A fist bump? A pound? A terrorist fist jab? The gesture everyone seems to interpret differently. We'll show you some interesting body communication and find out what it really says."
The gesture, derived from black street culture, is however commonplace in many walks of American life. Media commentators noted that former president George Bush once used it with the tennis player Anna Kournikova.
Ms Hill's remarks are one of several instances where the Right-wing network has disparaged the Obamas recently.
It characterised Mrs Obama as "Obama's Baby Mama", a derogatory term used for black single mothers, while Liz Trotta, a journalist and Fox contributor, joked about assassinating both Mr Obama and Osama bin Laden after supposedly muddling their names.
Both Ms Trotta and Ms Hill apologised, with the latter claiming that her comment had merely "characterised" discussions in the media about the Obamas' greeting.
However it appeared her analysis was extrapolated from a single comment on an article on the Right-wing HumanEvents.com, which posited that Hizbollah fighters used the same mode of greeting as the Obamas.
Some US commentators have noted that the Fox owner Rupert Murdoch spoke approvingly of Mr Obama recently, but it is not clear if the media mogul played any role demoting Ms Hill, who is however staying with the company. A Fox representative said changes to the afternoon schedule had been on the cards for weeks.
Karl Frisch, a spokesman for Media Matters, a liberal watchdog, said: "This is part of a broader problem with Fox: out of bound comments are followed by a half-baked apology. At some point Fox has got to decide if it is a responsible news source."
One of the first items on the new Obama website said that Mrs Obama, who like her husband is black, has never used the racially divisive term "whitey," as some blogs and conservative commentators have claimed.
It also tackles false claims that Obama, who will take on Republican Senator John McCain in the November election, follows Islam, and shows a photograph of Mr Obama with his hand on a Bible when he was sworn into the US Senate in 2007, to counter rumours that he used the Koran.