President Donald Trump, after reaching the White House with fiery rhetorical attacks and a combative message, pitched his agenda to voters and Congress with language that was much more presidential and traditional in tone. He also issued a call for American renewal in stark contrast to the aggressively nationalist posture he outlined at his inauguration just over a month ago.

Mr. Trump opened his first address to a joint session of Congress with a call to unity in the wake of attacks on Indian nationals in Kansas and the desecration of Jewish cemeteries: “We are a country that stands united in condemning hate and evil in all its forms.”

His speech largely avoided his signature attacks on his adversaries and the political establishment. While he once again highlighted the challenges of violent crime in some urban communities and drew attention to crimes committed by illegal immigrants, he didn’t repeat his denunciation of “American carnage” in his inaugural address.

Although there were no major policy shifts—he called for reworking trade deals and cracking down on illegal immigration without using the controversial “America First’’ label to describe them—the edges of his most incendiary rhetoric were sanded off. It was the Trump doctrine with aspirational overtones

In shifting his tone, Mr. Trump signaled he was willing to set aside campaign-style rhetoric for a less divisive style more conducive to governing—at least for the moment.

He called for lower taxes on corporations and middle-class Americans, for replacing the Affordable Care Act, for rebuilding the military and enforcing border security. Those are all standard elements of Republican orthodoxy.

He ticked off a series of initiatives that he believed could draw bipartisan support, such as making child care more accessible, ensuring paid family leave, supporting women’s health and rebuilding the nation’s infrastructure.

But despite the more unifying tone, the speech isn’t likely to do much to help Mr. Trump get his stalled legislative agenda off the starting block. To move a tax overhaul through Congress, and a replacement for the Affordable Care Act, he needs a united GOP and support from at least some Senate Democrats. Right now he has neither. His long speech isn’t likely to change that.

The ceremonial setting of Tuesday’s speech was emblematic of the pivot Mr. Trump faces at this point in his presidency. Delivering a formal address in the House chamber packed with the Washington establishment—including both friends and foes—was for Mr. Trump a far cry from the open-air campaign rallies and Inauguration Day address in which he fed off the energy of vast, admiring crowds.

Mr. Trump made no real concessions to attract bipartisan support to his policy platform. His signature health and tax policy proposals are stalled in Congress, with his own party at odds. His immigration policy has been stymied in court. His first month as president has been roiled by staff turmoil and intrigue.

He included in his list of bipartisan issues measures to “promote clean air and clear water,” but his administration has been moving to roll back antipollution regulations. He called for rebuilding the military, without mentioning that his proposed increases in the Pentagon budget would be accompanied by deep cuts in domestic spending.

He also signaled a willingness to negotiate on one of his signature issues: immigration.

“I believe that real and positive immigration reform is possible, as long as we focus on the following goals: To improve jobs and wages for Americans, to strengthen our nation’s security and to restore respect for our laws,” he said, although he didn’t commit to considering the lightning-rod issue of granting legal status to some people in the U.S. illegally. Democrats are sure to demand that a legalization plan be part of any package.

The speech came at a time of high risk for Mr. Trump’s agenda. Republicans are divided and looking to Mr. Trump for leadership on his plans to repeal and replace Obamacare and to revamp the tax code.

If Mr. Trump fails to deliver on those signature issues, it will undercut a crucial element of his political brand: He campaigned on the promise that, as a successful businessman, he would be a problem-solving, no-nonsense deal-cutter who would succeed where traditional politicians had failed.

The latest Wall Street Journal/NBC News poll suggests that for all the turmoil of the first month of the Trump presidency, many voters admire those qualities in him. A plurality of 46% rate him highly for his ability to “get things done,” and a plurality of 48% say he will bring “the right kind of change to the direction of the country.”

A key question is whether any Democrats in the Senate are open-minded enough to support any element of his agenda. He will need at least eight of them to move most legislation through the chamber. But many Democrats are under heavy pressure from the party’s liberal, activist base to oppose Mr. Trump at every turn.

His best hope is to draw support from 10 Senate Democrats up for reelection in 2018 in states that Mr. Trump won in 2016.

“There are a lot of Democrats who will do nothing but gain by being part of the unbending resistance to Trump, but there are others who stand to lose,” said William Galston, a former White House aide to President Bill Clinton who is now at the Brookings Institution.