Vice President Kamala Harris immediately pushed back on the idea that President Joe Biden should step aside, after Democratic Rep. Lloyd Doggett of Texas called on him to leave the race in the wake of his unsteady debate performance last week.

"Look, Joe Biden is our nominee," she said in an exclusive interview with CBS News. "We beat Trump once, and we're going to beat him again, period."

Harris said she's "proud to be Joe Biden's running mate" and declined to directly answer a question about whether she's ready to lead the country if necessary.

Speaking with CBS News' Nidia Cavazos after a fundraiser in San Francisco, Harris' remarks came as Democrats voice their concern about the president's fitness for another four years of the grueling job of the presidency and his ability to defeat former President Donald Trump. 

Doggett on Tuesday became the first Democratic lawmaker to call on Mr. Biden to step away from the race, and another group of lawmakers is also pleading with him to end his candidacy. Cavazos asked Harris for her response to Doggett and other concerned Democrats. 

"Are you ready to lead the country if necessary," Cavazos asked.

"I am proud to be Joe Biden's running mate," Harris responded.

The vice president has continued to hit the campaign trail while the president's team works to persuade voters that he's still the best choice. For his part, Mr. Biden is also raising money on Tuesday closer to home, in McLean, Virginia, where he apologized for his poor debate.

For the first time, he blamed his performance on his extensive foreign travel in early June.  

"I wasn't very smart," he told donors, claiming to have been in at least 15 time zones. He said he should have listened to his staff, who warned him it may not have been a good idea, and he said he nearly fell asleep on stage during the 90-minute debate.

The president and vice president have kept up their regular cadence of communication since the debate, according to a source familiar with their communications.

As the sitting vice president, Harris' name has been floated as one obvious potential option to replace Mr. Biden, if he were to step down. But that's something he would have to do voluntarily. Democratic Party officials can't force him off the ticket, now that primary voters have spoken. 

On Tuesday, Harris also commented on the Supreme Court's opinion Monday that Trump enjoys presidential immunity for his official acts as president. 

"It's one of the foundational principles of our system of justice that no one is above the law," Harris said. 

"When we have a candidate in Donald Trump that's openly said he'll be a dictator on Day One, that he will weaponize the Department of Justice against his political enemies, it is very likely that he could be immune from those kinds of acts, and we have to take seriously the stakes of this election in terms of the thought that we could have a president in the White House who thinks he's immune and able then to make decisions with that office that include weaponizing the Department of Justice against his political enemies." 

In its opinion, the Supreme Court did not grant former presidents absolute immunity from prosecution, but rather, divided presidential conduct into three categories regarding immunity: official acts that are part of presidents' "core constitutional powers"; other official acts that are outside their "exclusive authority"; and unofficial acts. Presidents have "absolute" immunity for the first category, "presumptive" immunity for the second and no immunity for the third. 

The high court offered an example of presumptive immunity in its opinion, citing prosecutors' contentions that Trump pressured his vice president, Mike Pence, to delay the certification of the Electoral College votes on Jan. 6, 2021, as he presided over the joint session of Congress. The special counsel will now likely have to "rebut the presumption of immunity" to show that Trump is not entitled to legal protection for this conduct.

Special counsel Jack Smith now bears the burden of proving that prosecuting Trump for allegedly pressuring Pence would not "pose any dangers of intrusion on the authority and functions of the Executive Branch." D.C. District Judge Tanya Chutkan will then have to make a determination on the matter.

The court also pointed to "a broad range of conduct" that the lower court will have to examine, including Smith's claims that Trump worked with state officials, private attorneys and his supporters outside the Capitol to subvert the transfer of presidential power.

Melissa Quinn, Robert Legare, Jenna Gibson and Josh Gross contributed to this report.

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• Supreme Court of the United States
• Joe Biden
• Donald Trump