"The behavior on the part of Russians, throughout the region and the world, is not acceptable", McCain, one of the strongest supporters of North Atlantic Treaty Organisation expansion in Europe, said in Montenegro.
China usually sides with Russian Federation in the Security Council, so the move to abstain represented a significant shift for Beijing.
Trump says he's "very impressed" with Xi, adding he thought he means well and wants to help.
Tillerson's trip has been dominated by US criticism of Russia's unwavering support for Syrian President Bashar Assad following a gruesome poison-gas attack in northern Syria last week blamed on Assad's forces.
Under Russian leader Vladimir Putin, the Kremlin has long seen the Balkans as inside its "sphere of influence".
He notes that Russians were operating on the Syrian air base from which the attack was launched last week.
Former CIA Director and Defense Secretary Leon Panetta told "CBS This Morning" on Tuesday that he doesn't believe that Bashar al-Assad has won Syria's civil war.
Matthew Rycroft told the Security Council: "The United Kingdom therefore shares the U.S. assessment that it is highly likely that the regime was responsible for a sarin attack on Khan Sheikhoun on 4 April".
Twitter challenges US order for anti-Trump user records
On Thursday, Twitter charged that efforts by the government to "unmask" the people behind the account violated the First Amendment.
Trump commented at a White House news conference with NATO Secretary-General Jens Stoltenberg (yehnz STOHL'-tehn-burg).
The United States does not want to see escalation in relations with Russian Federation and looks for a dialogue and cooperation, a senior White House official said in a briefing on Wednesday. Trump had previously questioned NATO's relevance.
He hailed NATO's role in the fight against terrorism, describing the alliance as the "bulwark of global peace and security".
"I have said it was obsolete".
Vice President Mike Pence met with Stoltenberg shortly after, reassuring the alliance that the US would meet its treaty obligations if necessary.
Ahead of President Trump's meeting with NATO Secretary General Jens Stoltenberg in Washington, senior White House officials said the USA supports Montenegro's investigation into the alleged Russian-backed plot to thwart the Balkan nation from joining the western security alliance.
Trump called Tillerson's work in Moscow "absolutely terrific" but said "the end result is what's important, not just talk".
Germany, for instance, has committed to ratcheting up its spending to 2 percent of GDP by 2024, a decision Trump praised Merkel for during a joint press conference at the White House last month. Just the USA and a handful of other countries are meeting the target.