President Barack Obama has inaugurated his second term with an ardent call for unity, but warned his foes their "absolutism" must not thwart action on climate, immigration and gun control.

Obama was publicly sworn in for another four White House years before a flag-waving crowd of an estimated one million, and then delivered an inaugural address in which poetic power veiled clear signs of a liberal governing agenda.

"We do not believe that in this country freedom is reserved for the lucky, or happiness for the few," Obama said Monday, from the Stars and Stripes-draped West Front of the US Capitol building, the epicenter of America's political divides.

The 44th president repeatedly used the "We the People" preamble to the US Constitution to suggest how to reconcile America's founding truths and the current discord and dysfunction of its embittered political system.

"Decisions are upon us, and we cannot afford delay," said Obama, flexing the freedom of a leader who no longer needs to face voters, and the urgency of a president who knows that second-term powers soon wane.

"We cannot mistake absolutism for principle, or substitute spectacle for politics, or treat name-calling as reasoned debate. We must act, knowing that our work will be imperfect."

Obama, America's first black president, took the oath of office with his hand resting on Bibles that once belonged to Abraham Lincoln, the president who ended slavery, and civil rights icon Martin Luther King, Jr.

He later signed King's Bible at his family's request.

In another echo of history, the inaugural ceremonies took place on the federal holiday marking King's birthday, and a majority of the crowd appeared to be African Americans.

Later, on a day mixing the pageantry of the presidency with the blunt business of drawing political battle lines, Obama and wife Michelle jumped out of their armored car to greet wildly cheering crowds in the inaugural parade.

And in the evening, the Obamas danced at both inaugural balls at the massive Washington Convention Center. The first lady, whose fashion choices are closely scrutinized, wore a red halter-neck gown by Jason Wu, who also designed her 2009 ball gown.

Though his speech was watched across the globe, Obama sketched over foreign policy, disdaining "perpetual war" and promising diplomatic engagement backed with military steel -- without dwelling on specific crises like Iran.

"We will show the courage to try and resolve our differences with other nations peacefully -- not because we are naive about the dangers we face, but because engagement can more durably lift suspicion and fear," he said.

While reaching for a soaring note of national unity, Obama's address was laced with liberal ideology, and policy certain to enrage Republicans.

"Together, we resolved that a great nation must care for the vulnerable, and protect its people from life's worst hazards and misfortune," the president said.

"Now, more than ever, we must do these things together, as one nation and one people."

While irking Republicans, Obama's signs of intent on issues like gun control and climate change may also worry Democrats from conservative territory running in 2014 mid-term polls who may hold the fate of his agenda in their hands.

In an apparent bid to frame his legacy, Obama said America must shield the weak, the poor and those lacking health care and demanded equality for all races and gay rights, and security from gun crime for children.

"Our journey is not complete until our gay brothers and sisters are treated like anyone else under the law -- for if we are truly created equal, then surely the love we commit to one another must be equal as well," Obama said.

"Our journey is not complete until we find a better way to welcome the striving, hopeful immigrants who still see America as a land of opportunity," he added, signaling a policy drive on a deeply contentious issue.

And in an oblique reference to his bid to end gun violence, Obama said: "our journey is not complete until all our children, from the streets of Detroit to the hills of Appalachia to the quiet lanes of Newtown, know that they are cared for, and cherished, and always safe from harm."

He also vowed to meet the threat of global warming, despite skepticism on climate change among some Republicans and daunting political and economic barriers to meaningful action.

Obama's Republican foes welcomed his reach for unity but, like the president, hinted at ideological divides.

"The president's second term represents a fresh start when it comes to dealing with the great challenges of our day; particularly, the transcendent challenge of unsustainable federal spending and debt," said Republican Senate Minority Leader Mitch McConnell.

Defeated Republican vice presidential candidate Paul Ryan congratulated Obama saying, "we serve the same country, one that is still in need of repair."

Speaking to AFP, Republican Senator John McCain, who lost the White House race to Obama in 2008, damned the address with faint praise.

"I thought it was an excellent speech, delivery was obviously excellent," McCain said. "I didn't hear any conciliatory remarks associated with it."

After his speech, Obama dined on bison and lobster with VIP members of Congress before heading back to the White House on the inaugural parade route.

Dozens of groups -- marching bands, cultural organizations and military units -- strode past the bullet-proof reviewing stand, earning applause from Obama, Vice President Joe Biden and other dignitaries.

Obama took the oath for a first time Sunday in a private ceremony at the White House because the Constitution states that US presidential terms end at noon on January 20.