Murdoch said the cartoon was intended only to "mock a badly written piece of legislation."
The cartoon, which was published Wednesday, depicted the body of the bullet-riddled chimp Travis and two police officers. The caption said: "They'll have to find someone else to write the next stimulus bill."
The chimp was killed in Connecticut last week after mauling a woman.
The president of the National Association for the Advancement of Colored People, which called for a boycott of the Post on Saturday, said Murdoch's pledge to be more sensitive was welcome but did not go far enough.
"Mr. Murdoch's apology comes only after almost a week of tens of thousands of expressions of outrage and disgust from people across the country," NAACP President Benjamin Todd Jealous said in a statement. "The offenders are still on staff and there are no measures being taken to increase diversity in its newsroom."
A spokesman for the Post said the paper was declining to comment on the NAACP statement or questions about whether anyone would be disciplined.
The Post also apologized Thursday in an online editorial.
The Rev. Al Sharpton is urging the Federal Communications Commission to review policies allowing Post owner News Corp. to control multiple media outlets in the same market.
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