President Trump scored a series of successes on his trip to Europe for the Group of 20 summit, including finalizing a regional cease-fire in Syria and confronting Russian President Vladimir Putin, but he couldn’t escape the dispute over Russian interference in the November election that clouds every accomplishment.

On his first day back in the White House on Sunday, Mr. Trump felt compelled to respond to his critics.

In six Twitter posts, the president defended his performance in his first face-to-face meeting with Mr. Putin and slammed former President Barack Obama and the Democratic National Committee for failing to prevent election meddling.

“I strongly pressed President Putin twice about Russian meddling in our election. He vehemently denied it. I’ve already given my opinion,” Mr. Trump tweeted. “We negotiated a cease-fire in parts of Syria which will save lives. Now it is time to move forward in working constructively with Russia!”

He touted an agreement with Russia to form “an impenetrable Cyber Security unit” to combat hacking and stressed that he never discussed the U.S. sanctions imposed against Moscow.

The Twitter tirade highlighted how Moscow’s interference in the presidential election and the left’s unsubstantiated accusations of Trump campaign collusion with Russians consume the news media and the Trump administration.

Senate Minority Leader Charles E. Schumer, New York Democrat, said that despite a few accomplishments coming out of the summit, Mr. Trump overall proved an embarrassment.

“Clearly the lowest moment of all was his meeting with Vladimir Putin, on several counts,” Mr. Schumer said. “Rather than decisively confronting the Russian president head-on, the president seemed to acquiesce to Putin’s denial, almost certainly paving the way for future Russian interference in our elections.”

As Mr. Trump flew back to Washington on Saturday, The New York Times reported about a previously undisclosed meeting last year at Trump Tower between the president’s eldest son, Donald Trump Jr., and a lawyer with ties to the Kremlin.

The meeting in June 2016 with the Russian lawyer, who was later identified as Natalia Veselnitskaya, was also attended by Mr. Trump’s son-in-law and top adviser Jared Kushner and Trump campaign chairman Paul J. Manafort.

Donald Trump Jr. acknowledged the meeting in a statement Saturday, saying the discussions with the lawyer primarily involved a program for U.S. adoptions of Russian children that was once popular but had been discontinued.

However, the younger Mr. Trump said Sunday that he had been told Ms. Veselnitskaya might have information that would be useful to the campaign.

He said that during the meeting Ms. Veselnitskaya claimed to have information that “individuals connected to Russia were funding the Democratic National Committee” and helping Mrs. Clinton.

“No details or supporting information was provided or even offered,” the younger Mr. Trump said in a statement. “It quickly became clear that she had no meaningful information.”

He said his father was unaware of the meeting.

The president’s opponents in the U.S. seized on the story as fresh evidence of possible collusion. The FBI has uncovered no evidence of collusion after a yearlong investigation, but investigations continues, including a Justice Department special counsel probe that began in May.

Still, Mr. Trump enjoyed a largely successful second trip abroad as president.

He delivered a speech in Warsaw, Poland, to a crowd of thousands that celebrated the West and called for an aggressive defense of its values of democracy and freedom that are under siege. The speech was widely praised as “Reaganesque.”

At the Group of 20 summit in Hamburg, Germany, Mr. Trump helped launch the World Bank’s Women Entrepreneurs Finance Initiative and announced a $50 million U.S. donation to finance the center.

He built support with Chinese President Xi Jinping for action to confront nuclear-armed and volatile North Korea after its successful test launch of a long-range missile capable of hitting Alaska.

The Trump administration also announced a cease-fire agreement for southwest Syria brokered by the U.S., Russia and Jordan.

The deal, finalized as Mr. Trump and Mr. Putin met for the first time, demonstrated a new level of U.S. involvement in ending Syria’s 6-year-old civil war in which the U.S. and Russia support opposing sides. The cease-fire took effect Sunday at noon Damascus time.

But it was questions about Russian election meddling and the relationship between Mr. Trump and Mr. Putin that dominated headlines.

Mr. Trump brought up the question of Russian interference in the U.S. presidential election at the top of the meeting with Mr. Putin on the sidelines of the G-20 summit. Mr. Putin reiterated his denial that Russia was involved.

The Russian president later fueled the controversy by saying he thought Mr. Trump accepted his denial.

“He asked many questions on that subject. I answered those questions as best I could,” Mr. Putin said. “I think he took it into consideration and agreed with me, but you should really ask him how he feels about it.”

White House Chief of Staff Reince Priebus rejected Russia’s characterization.

“It’s not true. The president absolutely did not believe the denial of President Putin,” Mr. Priebus told “Fox News Sunday.”

Mr. Trump has been criticized for expressing doubt about the source of interference in the election, although several U.S. intelligence agencies believe Russia was solely responsible. Mr. Trump said Thursday that he believed Russia was involved but that other countries or individuals also could have been involved.

Mr. Priebus reiterated that position.

“He said they probably meddled in the election. They did meddle in the election,” Mr. Priebus said. “The one thing that he also says, which drives the media crazy, but it’s an absolute fact, is that others have as well, and that’s true.”

The interference included the spread of disinformation during the campaign. The most prominent meddling, however, was hacking into emails at the Democratic National Committee and Democratic presidential nominee Hillary Clinton’s campaign officials, exposing embarrassing information.

There has been no evidence that the efforts impacted the outcome of the election, but Democrats have pointed to Russian interference to delegitimize the Trump presidency.

Even some Republicans fear that Mr. Trump has a blind spot when it comes to Russia despite taking tough stances against North Korea and the Islamic State terrorist group.

“To forgive and forget when it comes to Putin regarding cyberattacks is to empower Putin, and that’s exactly what he’s doing,” Sen. Lindsey Graham, South Carolina Republican, told NBC’s “Meet the Press.”

He ridiculed Mr. Trump’s tweet about working with Russia on cybersecurity, saying it is “not the dumbest idea I’ve ever heard, but it’s pretty close.”

Rep. Adam B. Schiff of California said, “We might as well just mail our ballot boxes to Moscow.”

“I don’t think that’s an answer at all,” Mr. Schiff, the top Democrat on the House Permanent Select Committee on Intelligence, told CNN’s “State of the Union.”

On Sunday evening, Mr. Trump seemed to back off on the idea, taking to Twitter to say “the fact that President Putin and I discussed a Cyber Security unit doesn’t mean I think it can happen.

Mr. Trump, meanwhile, signaled he is moving on. Before launching into his defense and counterattack on Twitter, he said he was pressing ahead with his agenda on trade.

“The G 20 Summit was a great success for the U.S. — Explained that the U.S. must fix the many bad trade deals it has made. Will get done!” he said.

And his Sunday evening cybersecurity tweet went on tout the Syrian cease-fire worked out by the U.S. and Russia. “It can’t,” he said of the cybersecurity unit, “but a ceasefire can,& did!”

• Tom Howell Jr. contributed to this report.

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