The newspaper acknowledged that the cartoon published on Wednesday had drawn controversy because African-Americans and others saw it as a depiction of Obama.
"This most certainly was not its intent; to those who were offended by the image, we apologise," the paper said in an editorial on its website headlined "That Cartoon."
"It was meant to mock an ineptly written federal stimulus bill. Period," the paper said.
The cartoon of a policeman shooting an ape played on the real shooting of a pet chimpanzee in Connecticut this week. A police officer in the cartoon says, "They'll have to find someone else to write the next stimulus bill."
The cartoon ran a day after Obama signed into law the $787 billion (Dh2.9 trillion) economic stimulus that he had strongly promoted. Critics interpreted the cartoon's dead chimp as a reference to Obama, who became the first black president of the United States on January 20.
Demonstrators led by civil rights activist Al Sharpton chanted "End racism now!" outside the skyscraper headquarters of the newspaper's parent company in midtown Manhattan on Thursday. They called for the jailing of Rupert Murdoch, whose international media conglomerate News Corp owns the Post.
The newspaper initially defended the cartoon as a parody of Washington politics, but Sharpton said it exploited a potent image in the history of racism toward blacks.
"I guess they thought we were chimpanzees," Sharpton said. "They will find out we are lions."
Sharpton said in a statement on Thursday night that groups protesting the cartoon would go ahead with a previously scheduled rally outside the Post on Friday afternoon and decide on a response to the Post editorial.
He added that "though we think it is the right thing for them to apologise to those they offended, they seem to want to blame the offense on those of whom raised the issue, rather than take responsibility for what they did."
The Post said it was not apologising to all of its critics.
"There are some in the media and in public life who have had differences with The Post in the past – and they see the incident as an opportunity for payback. To them, no apology is due," the editorial said.
"Sometimes a cartoon is just a cartoon – even as the opportunists seek to make it something else," it said.
Critics said the racist message was clear.
"You would have to be in a time warp or in a whole other world not to know what that means," said demonstrator Charles Ashley, 25, a model who did not believe the cartoon was an innocent political joke.
Others said it made light of assassinating Obama, a possibility they said that worries many Americans.
"Just the fact that they put a monkey with gunshot wounds in his chest, it gives the idea of an assassination," said Peter Aviles, 48, a building superintendent.
Police in Stamford, Connecticut, shot and killed a 200-pound (90-kg) chimpanzee on Monday after the pet nearly killed its owner's friend and attacked a police car. The chimp, named Travis, had once starred in television commercials and was taking medication for Lyme disease.