Washington (CNN) -- President Barack Obama emerged Thursday from a meeting with Arizona Gov. Jan Brewer proclaiming limited progress on the immigration issue that divides them, but sticking to his opposition to her state's controversial immigration bill.

"I've told Governor Brewer that we've already put more resources into border security than we ever have," Obama told CNN's Larry King in an interview to air at 9 p.m. Thursday.

"We have got more border guards in Arizona than we ever have. We just made decisions to put in additional National Guards. But without comprehensive immigration reform -- that is Congress' responsibility -- we are not going to solve this problem and that's what we have to do."

Brewer, who signed the state's immigration law that Obama has called misguided, described the tone of the talks as "very cordial" but said disagreements remained. "We know we're not going to agree on some issues until other issues are worked out," the Republican governor told reporters.

Brewer said Obama agreed to send staffers to Arizona "in a couple of weeks" to discuss using more federal resources to tighten the border, including allocating 1,200 National Guard troops and $500 million more in funds for border protection and law enforcement.

"He assured us that the majority of those resources would be coming to Arizona," Brewer said.

She added, "I am encouraged there is going to be much better dialogue between the federal government and the state of Arizona. I hope that that's not wishful thinking; I hope that that's positive thinking."

But Obama did not appear to concede any ground on his opposition to the Arizona law.

"Although I understand the frustration of the people of Arizona when it comes to the inflow of illegal immigrants, I don't think this is the right way to do it. I think this puts American citizens who look Hispanic, are Hispanic, potentially in an unfair situation."

More importantly, he said, "it also creates the prospects of 50 different laws in 50 different states when it comes to immigration."

He said the problem would not be solved until Congress passes "comprehensive immigration reform."

Outside the White House, dozens of people protested the law. "Jan Brewer, shame on you!" they chanted. One man wore a T-shirt that said, "Shame on Arizona. Boycott Racism."

The meeting came after the governor said this week she is not worried about a potential legal challenge from the Obama administration over the law.

"We'll meet you in court," Brewer told CNN's "John King, USA," in a reference to the adminisration. "I have a pretty good record of winning in court."

The American Civil Liberties Union is leading a court challenge against the Arizona law.

U.S. Attorney General Eric Holder, who discussed the law last week with a delegation of police chiefs from Arizona, has yet to indicate whether the federal government will file a legal challenge and Obama refused comment Thursday.

"That's really the job of the Justice Department and I made a commitment early on that I wouldn't be putting my thumb on the scales when these kinds of decisions are made," the president said.

The law, which Brewer signed in April, would allow police officers to check the residency status of anyone who is being investigated for a crime or possible legal infraction if there is reasonable suspicion the person is an illegal resident.

Critics, including Holder, have said the law will promote racial profiling.

But Brewer said this week the law does not target an individual's specific race. She also said driver's licenses do not prove citizenship.

"It wouldn't matter if you are Latino or Hispanic or Norwegian," she said. "If you didn't have proof of citizenship and the police officer had reasonable suspicion, he would ask and verify your citizenship. I mean, that's the way that it is. That's what the federal law says. And that's what the law in Arizona says."

Brewer said she would not suspend the law even if Obama were to increase the number of U.S. troops at the Mexican border.

CNN's Alex Mooney and Ed Henry contributed to this report.