Another Republican senator, Markwayne Mullin of Oklahoma, went further and appeared to suggest that Mr. Biden should be forced from office by exercising the 25th Amendment of the US Constitution – a never-used method to replace the president if he cannot fulfill his duties.
Critics of Trump had called for using the amendment to remove him when he was in office.
Many in the political world had been expecting to Mr Biden to drop out of the race.
His rambling, frequently incoherent answers in the June 27 debate with Trump had stunned the country and left people wondering if he could serve as president for another four years. While in speeches and interviews Mr. Biden often showed renewed vigor, he was also dogged by major stumbles and seeming memory problems.
Democrats in Congress, worried that his shakiness would hurt their chances at re-election, and major donors began to press for him to drop out, but they did not press for him to resign.
The last president to abandon his election campaign, Lyndon B. Johnson, also served out the remainder of his term. Like Mr Biden, Johnson had said that giving up the race would allow him to focus on his presidential duties.
As the pressure on Mr Biden has grown in recent weeks, Republicans became more vocal about a resignation.
Just hours before the president announced he was stepping aside, Trump’s new running mate, Senator JD Vance of Ohio, said: “Not running for reelection would be a clear admission that President Trump was right all along about Biden not being mentally fit enough to serve as Commander-in-Chief. There is no middle ground.”
“Joe Biden has been the worst President in my lifetime and Kamala Harris has been right there with him every step of the way,” he added.
Mr Biden has endorsed Ms Harris to take up the mantle of the presidential campaign, although the party will still have to formally approve its nominee.
“I am honored to have the President’s endorsement and my intention is to earn and win this nomination,” Ms Harris said in a statement. “I will do everything in my power to unite the Democratic Party—and unite our nation—to defeat Donald Trump and his extreme Project 2025 agenda.”
Republicans have reportedly prepared to attack Ms Harris’s candidacy – as many believed she was the most likely successor.
Trump campaign sources have told US media outlets that they were readying attack ads and opposition research in case they faced her.
Most criticism centres on the vice-president’s lead role on immigration issues within the administration. Several speakers at the Republican convention last week portrayed Ms Harris as a failed “border czar”.
Those attacks returned on Sunday.
Speaker Johnson called her “a completely inept border czar” and said she had been “a gleeful accomplice” in “the destruction of American sovereignty, security, and prosperity”.
“She has known for as long as anyone of his incapacity to serve,” he said, while also accusing her of being part of a political coverup of Mr Biden’s problems.
Republican Governor Greg Abbott of Texas, known for taking a hard line on immigration that has led to legal actions, also expressed concerns about Ms Harris becoming president.
“I think I will need to triple the border wall, razor wire barriers and National Guard on the border,” he wrote on social media.
Donald Trump Jr, the former president’s son, broadly said her policies would be no different than Mr Biden’s.
“Kamala Harris owns the entire left-wing policy record of Joe Biden. The only difference is that she is even more liberal and less competent than Joe, which is really saying something,” he posted on X, formerly Twitter.