Putin, who was recently reelected to a third six-year term in what critics described as a "sham election", responded to Wallace by likening these suspicious deaths to the assassinations of USA presidents and civil rights leaders.
Earlier, Trump took to Twitter to blame America for deteriorating relations between Russia and the U.S. - a sentiment the Russian foreign minister quickly approved of, tweeting, "We agree". "His job is in representing his country and he is doing that to the best of his ability".
"They said they think it's Russian Federation".
Mr Trump addressed the fact that he thinks the countries "have not been getting along very well for the last number of years" but that "I really think the world wants to see us get along".
Separately, various US and foreign diplomats told McClatchy that Trump's remarks leave them anxious America will no longer be a reliable partner, will no longer defend other countries and will no longer stand up to its adversaries.
Mr Trump's team says "a productive dialogue" with Russian Federation is "good for the world".
Sen. John McCain of Arizona, who also has not seen eye-to-eye with the president, said the press conference "was one of the most disgraceful performances by an American president in memory". He said Mr. Trump's behavior "may reflect that concern" that the Russians will use his past against him. "If they instead contained damaging information - say, regarding Clinton Foundation fundraising - the new president would have taken office in the shadow of a sword dangling from a string held by the Russians".
The meeting in Helsinki took place in the midst of frayed relations between the U.S. and its European allies, following a contentious North Atlantic Treaty Organisation summit, and after 12 Russian military intelligence officers were indicted in the USA for hacking Democrats in the 2016 election.
During the news brief, which came after the leaders met privately for over two hours, Trump seemed to accept Putin's denial of interfering in the 2016 presidential election - despite findings to the contrary by USA intelligence agencies.
Lasha Tchantouridze is a professor of diplomacy and worldwide relations at Norwich University, who specializes in Russian Federation and the U.S. He says this meeting between the two leaders could be the start of something different.
"The president must appreciate that Russian Federation is not our ally", said Republican House Speaker Paul Ryan.
Trump's comments come after he demanded last week that North Atlantic Treaty Organisation allies "reimburse" the USA for defence costs.
House Speaker Paul Ryan, who rarely criticizes Trump, stressed there was "no question" that Russian Federation had interfered. The speaker said he stood by his Monday statement that "the president must appreciate that Russian Federation is not our ally".
David Axelrod, a former top adviser of President Barack Obama, condemned Trump's "treacherous performance", and called for a "thorough and complete" probe by special investigators and Congress. "What they did was a direct attack on our democracy".