However, Mr Trump said that Secretary of State Rex Tillerson's meetings in Moscow this week, which included a meeting with Russian President Vladimir Putin, went better than expected.
Russian Foreign Minister Sergey Lavrov, who hosted his Iranian and Syrian counterparts at a trilateral meeting in Moscow, denounced last week's US attack on Syria as a "flagrant violation" of worldwide law and warned that any further such action would entail "grave consequences not only for regional but global security".
Russia's foreign minister Sergei Lavrov had said his country agreed a United Nations investigation into the attack just hours before it vetoed the resolution, which also called for an investigation, on Wednesday. The missile strike, which targeted a Syrian military air base, followed a chemical weapon attack that the USA and other Western nations have blamed on Assad, an assessment that the Kremlin disputes.
Mr Tillerson also called for Assad to eventually relinquish power. Secretary of State Rex Tillerson said the US sees "no further role" for Bashar al-Assad as Syria's leader.
Russia's veto was the eighth by the country at the UN Security Council, in a bid to shield Syrian President Bashar al-Assad.
"Right now we're not getting along with Russian Federation at all", Mr. Trump said flatly during a White House news conference on Wednesday.
The Russian and Syrian governments have both denied involvement in the chemical attack.
Among 15 council members, ten voted in favor; Bolivia and the council's veto-wielding power Russian Federation voted against; China, Ethiopia, Kazakhstan abstained.
Tillerson and Lavrov said in a joint press conference that Syria was a major subject in their discussions.
The shape of these relations will also greatly influence the region, which watches with interest the spar between the two giants pumped up by too much ego and with too many interests vested to make them pliant.
Putin received Tillerson at the Kremlin along with Russia's Foreign Minister Sergei Lavrov after the top diplomats held several hours of talks dominated by the fallout of an alleged chemical attack in Syria. "And we have to find ways to live with them and to try and avoid a new Cold War, a new arms race", he said.
"It'd be a fantastic thing if we got along with Putin and if we got along with Russia", Mr. Trump said. Perhaps eager to take a shot at the previous US administration, with whom Putin's government clashed badly, Lavrov said the new group would address "irritants which have dogged our relations over the last couple of years, particularly under the administration of President Obama".
Secretary of State, Rex Tillerson met with Vladimir Putin in Russian Federation. "I like to think they didn't know".
Despite hopes of an improvement in ties under Trump, the Tillerson-Lavrov talks look set to be dominated by the war of words over Syria - where more than 320,000 people have died in six years of brutal war. "I feel ashamed, I feel ashamed".