United States President Joe Biden has said it was wrong to say Donald Trump should be put “in a bull’s-eye” but defended his portrayal of his Republican rival as a threat to democracy.

In his first television interview since Trump narrowly survived an attempted assassination, Biden said the comments he made in a private call with donors were intended to put the focus on his challenger’s policies and character.

“It was a mistake to use the word. I didn’t say ‘crosshairs’. I meant bulls-eye, focus on him, focus on what he is doing,” Biden said in an interview with NBC News’s Lester Holt on Monday.

“Focus on his policies, focus on the number of lies he told in the debate.”

Asked if he had done any “soul searching” about his rhetoric since the attempt on Trump’s life, Biden said it was appropriate to highlight the dangers posed by Trump and denied using inflammatory language like his rival.

“Look, how do you talk about the threat to democracy, which is real, when a president says things like he says? Do you just not say anything because it may incite somebody?” Biden said.

“Look, I have not engaged in that rhetoric. Now, my opponent has engaged in that rhetoric. He talks about, there’ll be a bloodbath if he loses.”

Biden’s comments came after several prominent Republicans, including Trump’s vice-presidential running mate DJ Vance, accused the president and his allies of creating the conditions for violence through their rhetoric casting Trump as an existential threat to democracy.

Biden, who has made the preservation of US democracy a key message of his re-election campaign, temporarily suspended television advertisements and an appearance in Texas following the attack, but will pick up his campaign with several events this week in the swing state of Nevada.

Trump came within centimetres of death on Saturday after a gunman opened fire on a campaign rally in Butler, Pennsylvania, striking the former president in his right ear.

Corey Comperatore, a 50-year-old former fire chief, was killed and several others were injured in the attack.

Authorities have yet to identify a motive for the suspected shooter, Thomas Matthew Crooks, 20, who was shot dead shortly after he opened fire.

The assassination attempt has prompted widespread calls for a reset of the US’s acrimonious politics, with both Biden and Trump calling on Americans to put political divisions aside and come together.

Biden on Sunday called on Americans to “lower the temperature in our politics,” while Trump said in a newspaper interview that he would rewrite his speech for the Republican National Committee to stress the need for unity.

Republican Party Chairman Michael Whatley on Monday continued the theme of unity at the opening of the RNC, telling supporters: “We must unite as a party, and we must unite as a nation.”

Trump made his first public appearance since the attack at the RNC later on Monday, receiving a rapturous reception from supporters as he entered the Fiserv Forum stadium to the tune of “God Bless the USA.”

While the effect of the attack on Trump on the race is not yet clear, some analysts have suggested that it is likely to bolster his standing in the polls, which already show Biden behind, both nationwide and in key battleground states.

In his NBC News interview, Biden, 81, reiterated his determination to stay in the race despite concerns about his age and fitness, which came to the fore after his poor debate performance against Trump last month.

In response to a question about whether he believed he had weathered calls for him to step down from within his own party, Biden said that 14 million Democratic voters had chosen him as their candidate.

“I listen to them,” Biden said.

Asked who he would listen to about the decision to stay in the race, Biden said himself.

“The idea I am the old guy – I am, I’m old. But I’m only three years older than Trump, number one,” he said.

“And number two, my mental acuity has been pretty damn good. I have gotten more done than any president has in a long, long time in three and a half years.”