Chris Christie: Carly 'rude' & Trump leads—for now

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Republican presidential candidate Chris Christie said Friday the GOP hopefuls should quit attacking each other and sharpen their assault on the record of President Barack Obama and Democratic hopeful Hillary Clinton.

Christie said Donald Trump is the frontrunner "for the moment," using the analogy Rick Perry in 2012 was at 30 percent in the presidential polls at the same point in the cycle, and four years later, he's out of the race for the second time.

The Trump factor in Wednesday night's debate gave CNN a network-record 22.9 million viewers, according to preliminary ratings. Fox drew an audience of 24 million last month. CNBC sponsors the next Republican debate on Oct. 28.

Politics is volatile business, Christie told CNBC's "Squawk Box" in an interview—adding that voters reward steady candidates with a serious message. "We know this will come around," he asserted.

The New Jersey governor also defended abruptly telling rival Carly Fiorina in the debate not to interrupt him.

While Christie received high marks for his performance, the former Hewlett-Packard CEO Fiorina was viewed by many observers as the candidate who stole the show.

Read More Who won the GOP's debate? Hint: The one in the dress

In one exchange, Christie said to Fiorina: "Carly, listen. You can interrupt anybody else on this stage, you can't interrupt me."

Asked about whether that was a fair reprimand, "I think she was being rude," Christie told CNBC Friday. "Carly is not going to be treated any differently by me on a debate stage because she's a woman And she wouldn't want it."

He pointed out that he spoke the same way to candidate John Kasich, governor of Ohio, but nobody raised any questions. "I treat women the same way that I treat men," he said. "I tell them the truth. And that's what they want to hear."

Christie also reacted to the firing of United Continental CEO Jeff Smisek amid an investigation into whether Smisek traded favors with the former chairman of the Port Authority of New York and New Jersey.

"I don't know anything about what went on there. Let's not jump to conclusions on things. I learned that when I was a prosecutor," he said. "Nobody knows whether anything went on. Yet in our 'accuse first and apologize later' society we indict, convict in the public medium before anything happens."

The inquiry grew out of the so-called Bridgegate scandal, which resulted in three former Christie aides being charged with engineering massive traffic gridlock in 2013 at the George Washington Bridge, run by the Port Authority, as political retaliation. The governor has not been implicated.

On CNBC Friday, Christie also talked about the economic progress he's made in New Jersey, which traditionally has been a Democratic state.

"We have a tough environment in New Jersey. I'm the first one to admit it," he said. "I wish we would lower taxes more. I vetoed more tax increases than any governor in American history."

He pointed to New Jersey's improving job market, which saw the state unemployment rate fall to a seven-year low of 5.7 percent. However, that's still higher than the national jobless rate of 5.1 percent.