The brief encounter -- out of earshot of observers but captured on camera -- was a highly visible demonstration of the verbal and legal skirmishing that has regularly occurred between Brewer and Obama's administration over illegal immigration and other issues.
Airport arrivals for presidents normally involve mere pleasantries between those involved, but Brewer and Obama have a history. And part of that history is what apparently got things going, according to accounts provided by Brewer and the White House.
Brewer said that during their talk, she invited Obama to visit Arizona to hear about her administration's achievements and to visit the U.S.-Mexico border, which has been a point of friction between the two because of illegal immigration issue.
Obama then said Brewer's recently published book mischaracterized a 2011 White House meeting between them.
Brewer said in an interview Thursday, at another Phoenix-area airport, that she talks a lot with her hands and that her pointing a finger at Obama during their conversation wasn't disrespectful.
"I respect the office of the president," she said. "I was there to welcome him."
She said she was grateful for the visit and intended to talk to him about the state's accomplishments. But she said she was "taken aback by his comments" when he said he wasn't happy with how her book described their White House meeting.
Immediately after the meeting Brewer had said it was cordial, but her book said Obama lectured Brewer in the Oval Office and that she felt he was condescending toward her.
"It is what it is. I proceeded to say that to him, and he chose to walk away from me," she said Thursday.
Asked whether she regarded that as disrespectful, she replied: "Well, I would never have walked away from anybody having a conversation. And, of course, that is what it is. It is disrespectful for me."
Their relationship covers disagreements on "most of his policies," she said. "That doesn't mean we can't be cordial to one another."
The encounter was notable because it was rare case of an unscripted and tense moment between the president and a public official in view of reporters.
White House press secretary Jay Carney chided reporters Thursday, saying the encounter with Brewer was getting too much attention from the press corps. The media coverage was overshadowing Obama's message of the day on energy.
Carney was questioned about Brewer's statement Thursday that Obama cut her short by walking away.
"You guys are giving this incident far too much importance, Carney said. He added, "I really assume you guys have more important issues to cover than this."
Brewer is among the Republican governors who oppose the federal health care overhaul, but the illegal immigration issue has been a particular sore point between Obama and Brewer.
The U.S. Justice Department has challenged Arizona's controversial 2010 immigration enforcement law in court, while the administration and Brewer are at odds over whether the federal government has done enough to secure the U.S.-Mexico border.
On Thursday, Brewer drew support from callers to conservative-oriented talk shows, but the incident left others in the state shaking their heads.
The Arizona Republic, the state's largest newspaper, editorialized that the image of Brewer wagging a scolding finger at the visiting president "now pretty much defines this state's relationship with Washington, D.C., to the world."
Bruce Merrill, a longtime Arizona pollster and a professor emeritus at Arizona State University, said there are two sides to the encounter, so it's hard to fully analyze what happened and why.
But the incident follows past incidents in which Arizona for a time balked at declaring a state holiday to honor Martin Luther King Jr., and Arizona State University refused to award Obama an honorary degree, Merrill noted.
"It reinforces the image of Arizona being kind of a cowboy state that doesn't show a lot of respect," he said of the airport encounter.
The two mayors who stood next to Brewer during the airport encounter were not available for interviews Thursday, their offices said.
"He doesn't want to get involved," said Melissa Randazzo, spokeswoman for Mesa Mayor Scott Smith, a Republican.
Phoenix Mayor Greg Stanton's office said his schedule had no time for an interview. Stanton is a Democrat.
Associated Press writer Jim Kuhnhenn contributed to this report from aboard Air Force One.