“What we can’t have is the same old politics of division that we have seen so many times before that dates back centuries,” Mr. Obama told a campaign rally for Philip D. Murphy in Newark. “Some of the politics we see now, we thought we put that to bed. That has folks looking 50 years back. It’s the 21st century, not the 19th century. Come on!”
Later, in the Virginia state capital of Richmond, he seemed to respond to Mr. Trump’s claim this week that Mr. Obama had not shown interest in the families of troops killed in combat. He did so by invoking the candidate Ralph S. Northam’s work as an Army doctor caring for veterans at Walter Reed National Military Medical Center.
“I can tell you as somebody who visited Walter Reed consistently throughout my eight years what it meant to have a medical staff that was literally helping to rebuild people’s lives,” Mr. Obama said.
Both former presidents have largely avoided taking on Mr. Trump since he was inaugurated in January, aside from occasional statements or comments in interviews. But the sight of the two most recent presidents back on the public stage on the same day, however coincidental, reinforced the broader alarm among establishment leaders of both parties.
“The two presidents speaking out so forcefully and eloquently is a warning that some basic principles of democracy that both parties have long supported at home and abroad are in jeopardy,” said Antony J. Blinken, who served as Mr. Obama’s deputy secretary of state and attended Mr. Bush’s speech on Thursday.
The bipartisan apprehension was illustrated by Mr. Blinken’s presence. As managing director of the newly formed Penn Biden Center for Diplomacy and Global Engagement led by Mr. Obama’s vice president, Joseph R. Biden Jr., Mr. Blinken attended to kick off a joint project with the George W. Bush Institute and Freedom House to counter the erosion of support for democratic principles and institutions at home and abroad.
Similarly, former Secretary of State Madeleine K. Albright, who served under President Bill Clinton, joined former Secretary of State Condoleezza Rice, who served under Mr. Bush, for a panel discussion with Nikki R. Haley, the ambassador to the United Nations. At times, the two former secretaries gently coached Ms. Haley to resist Mr. Trump’s efforts to cut the State Department budget.
Afterward, Mr. Bush and Ms. Albright hugged and sat together, with the former president draping his arm over her shoulders.
Mr. Bush also released a “call to action” report examining threats to the liberal democratic order and making recommendations for protecting American institutions. The paper was drafted by Peter H. Wehner, a former adviser in his White House, and Thomas O. Melia, a former State Department official under Mr. Obama.
For Mr. Bush, democracy and free trade are longtime themes, but there was an edge in his address that went beyond the usual nostrums. Asked by a reporter as he left the hall whether his message would be heard in the White House, Mr. Bush smiled, nodded slightly and said, “I think it will.”
In his speech, the former president lamented that “bigotry seems emboldened” and “our politics seems more vulnerable to conspiracy theories and outright fabrication.”
Pointing a finger at the nation’s leaders, he said, “We know that when we lose sight of our ideals, it is not democracy that has failed; it the failure of those charged with preserving and protecting democracy.”
He acknowledged public discontent. “We should not be blind to the economic and social dislocations caused by globalization,” he said. “People are hurting. They’re angry and they’re frustrated. We must hear and help them. But we cannot wish globalization away any more than we could wish away the agricultural revolution or the industrial revolution.”
He also offered what seemed like a rejoinder to a president who uses Twitter as a weapon in a perpetual political war. “Bullying and prejudice in our public life sets a national tone, provides permission for cruelty and bigotry and compromises the moral education of children,” Mr. Bush said.
The Bush family has never been fond of Mr. Trump, who belittled Jeb Bush during their contest for the Republican presidential nomination last year. Neither the former president nor his father, former President George Bush, voted for Mr. Trump, and the two issued a joint statement in August denouncing white supremacists after the violence in Charlottesville, Va., which Mr. Trump blamed on “both sides.”
The younger Mr. Bush seemed to return to that on Thursday. “Bigotry or white supremacy in any form is blasphemy against the American creed,” he said.
He also emphasized the seriousness of the Russian effort to influence last year’s election, interference that Mr. Trump has dismissed as a “hoax” perpetuated by Democrats and the news media. “America has experienced a sustained attempt by a hostile power to feed and exploit our country’s divisions,” Mr. Bush said.
Advisers and allies to Mr. Bush said he spoke out because he was troubled about the larger forces he sees in the United States and around the world. Tom Bernstein, a longtime friend, said that the moment was a “stress test for democracy” and that Mr. Bush wanted to make his points in a “very direct but very dignified” way.
“We’re all called on to make sure that we get our country back, and I think all the things the president spoke of today, it’s a reaffirmation of American values,” Mr. Bernstein said.
Mr. Wehner said Mr. Bush was not interested in quarreling with Mr. Trump. “There’s enough political food fighting going on,” he said. “He doesn’t want to be part of that. It’s not part of that. What we need is people with some authority in American life to articulate a vision of the common good and the moral good and a vision of America.”
Emulating Mr. Bush, Mr. Obama has mostly stayed quiet since leaving office. But with accomplishments like his health care program under siege, he returned to the fray at least elliptically on Thursday.
“You notice I haven’t been commenting a lot on politics lately,” he said in Richmond. “But here’s one thing I know: If you have to win a campaign by dividing people, you’re not going be able to govern them.”
He added: “Instead of looking for ways to work together to get things done in a practical way, we’ve got folks who are deliberately trying to make folks angry, to demonize people who have different ideas, to get the base all riled up because it provides a short-term tactical advantage.”
Mr. Obama was energized and comfortable, largely steering away from specific policy debates, but he made a point of noting in Newark that he “created millions of job” and “by the way, we covered a whole bunch of folks with insurance, too.” His allusions to Mr. Trump, however, were still clear.
“The world counts on America having its act together,” he said. “The world is looking to us as an example. The world asks what our values and ideals are and are we living up to our creed.”
An earlier version of a picture caption with this article misspelled the given name of a former secretary of state. She is Madeleine K. Albright, not Madeline.