He said the tape-recorded comments "made me cringe" when he read them in Friday's Los Angeles Times.
"Anyone out there that feels offended by those comments, I just want to say I'm sorry, I apologize," Schwarzenegger said. He added that if he heard his children make similar comments, "I would be upset."
The statements were captured on a six-minute tape made during a March 3 speechwriting session between Schwarzenegger and his advisers. On it, Schwarzenegger and chief of staff Susan Kennedy speak affectionately of state Assemblywoman Bonnie Garcia and speculate about her nationality.
"I mean Cuban, Puerto-Rican, they are all very hot," the governor says on the recording. "They have the, you know, part of the black blood in them and part of the Latino blood in them that together makes it."
Garcia, who is Puerto Rican, appeared with Schwarzenegger on Friday and said she was not offended by the governor's comments. Garcia earlier told the Times that she often calls herself a "hot-blooded Latina."
Schwarzenegger also said he called leaders from ethnic groups, who he said were not upset.
"All of them understood it was an off-the-record conversation," said Schwarzenegger, a Republican running for re-election in November. "It was not meant to be in any negative way."
A spokesman for Democratic Assembly Speaker Fabian Nunez declined to comment directly on the remark but said the governor "has always been very respectful toward Latinos."
"These are hardly Nixon's Watergate tapes," Nunez spokesman Richard Stapler said.
However, Schwarzenegger's Democratic challenger, Phil Angelides, said the governor should "conduct himself with dignity."
"Once again, Gov. Schwarzenegger has used language that is deeply offensive to all Californians and embarrassed our state," Angelides said in a statement.
Schwarzenegger aides routinely tape his speechwriting sessions so the writers can keep a record of his thoughts and speaking patterns.
The newspaper did not say how the tape was obtained. The participants suggest during the meeting that they know they are being recorded.