During the conference, Trump claimed that he believed that Russian Federation did nothing to interfere with the election because of Putin's denial.
There were indications of an arrangement to work together and with Israel to support a ceasefire in southern Syria, suggesting that the U.S. administration is backing off its demand that Moscow's ally Bashar al-Assad step down.
The senator expressed his concerns with Trump aligning himself too closely with Putin, citing his many alleged crimes.
Standing alongside the Kremlin boss at a joint news conference, Trump acknowledged that his intelligence chiefs believe Russian Federation hacked and leaked Democrats' emails containing politically damaging information about his rival Hillary Clinton in 2016.
Mr Trump pushed back, tweeting on Tuesday: "While I had a great meeting with North Atlantic Treaty Organisation..." On Monday, officials said another Russian agent had been arrested for seeking to influence U.S. politics. Instead, he directed his ire at Democrats and USA officials, calling special counsel Robert Mueller's probe of Russian Federation a "disaster". I think we've all been foolish.
Still, Putin said he had indeed wanted Trump to win the election - a revelation that might have made more headlines if not for Trump's performance - but had taken no action to make it happen.
Rep. Darrell Issa of California he takes the charges filed by Mueller's team seriously, but added, "I personally would neither rule in nor rule out the validity of a very interesting and odd-timed indictment of people who can never be brought to justice".
"However, I also recognize that in order to build a brighter future, we can not exclusively focus on the past - as the world's two largest nuclear powers, we must get along".
Former CIA Director John Brennan denounced Trump's performance as "treasonous", and Republican US Senator John McCain called the meeting with Putin a "tragic mistake", although some other Republicans were more cautious.
Asked if he believed United States intelligence agencies, which concluded that Russian Federation interfered in the 2016 election in an effort to help him defeat Democrat Hillary Clinton, Trump said he was not convinced it was Moscow.
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Russia's World Cup organizers said on Saturday that the tournament had so far been a success and would help to improve the country's image overseas .
But the top Democrat in the US Senate, Chuck Schumer, tweeted that many Americans can only wonder if "the only possible explanation for this risky behaviour is the possibility that President Putin holds damaging information over President Trump". John Brennan, who served as Central Intelligence Agency director, called Trump's comments "treasonous".
The pair had held lengthy talks before - on the sidelines of world leader meetings in Germany and Vietnam previous year - but this was their first official summit and was being watched closely, especially following the announcement Friday of new indictments against 12 Russian intelligence officers accused of hacking Democratic emails to help Trump's campaign.
"Please get this rubbish out of your heads", the Russian leader said.
"Speaking about having the ball in our court in Syria", Putin said, in an awkward change of subject.
Bent on forging a personal bond with the Kremlin chief, Mr Trump headed into the summit blaming the "stupidity" of his predecessors for plunging ties to their present low.
The summit capped a trip overseas during which Trump accused North Atlantic Treaty Organisation allies of failing to spend enough on their militaries and embarrassed British Prime Minister Theresa May by saying she refused to take his advice about how to negotiate Britain's exit from the EU.
A post-NATO trip to Britain, supposedly America's partner in a "special relationship", was riddled with controversy as well. Brennan tweeted: "Not only were Trump's comments imbecilic, he is wholly in the pocket of Putin".
Trump began the day by firing a Twitter broadside at his domestic opponents, blaming the diplomatic chill on the election investigation.
Rep. Liz Cheney, R-Wyo., the daughter of former Vice President Dick Cheney, said she is "deeply troubled" by Trump's defense of Putin against USA intelligence agencies "and his suggestion of moral equivalence" between the two countries.
Russia's foreign ministry tweeted in response: "We agree". He referred to the European Union as a "foe" in trade and repeatedly criticized it.