* China rebuffs Trump's criticism on Twitter
* Russia says attempts to shift responsibility to Moscow, Beijing 'baseless'
* Japanese PM Abe speaks with Trump after latest N. Korea test
* U.S. Ambassador to UN: China must decide on stronger sanctions
* North Korea poses "grave and direct threat" - White House (Adds apparent break-up of re-entry vehicle, paragraph 4)
By Ben Blanchard and Elias Glenn
BEIJING, July 31 (Reuters) - China hit back on Monday after U.S. President Donald Trump tweeted he was "very disappointed" in China following North Korea's latest missile test, saying the problem did not arise in China and that all sides need to work for a solution.
China has become increasingly frustrated with American and Japanese criticism that it should do more to rein in Pyongyang. China is North Korea's closest ally, but Beijing, too, is angry with its continued nuclear and missile tests.
North Korea said on Saturday it had conducted another successful test of an intercontinental ballistic missile that proved its ability to strike the U.S. mainland, drawing a sharp warning from Trump and a rebuke from China.
Video of the latest missile test appears to show it breaking up before landing, indicating Pyongyang may not yet have mastered re-entry technology needed for an operational nuclear-tipped missile, a think tank reported on Monday.
Japanese Prime Minister Shinzo Abe spoke with Trump on Monday and agreed on the need for more action on North Korea just hours after the U.S. Ambassador to the United Nations said Washington is "done talking about North Korea".
A White House statement after the phone call said the two leaders "agreed that North Korea poses a grave and growing direct threat to the United States, Japan, the Republic of Korea, and other countries near and far".
It said Trump "reaffirmed our ironclad commitment" to defend Japan and South Korea from any attack, "using the full range of United States capabilities".
Trump tweeted on Saturday after the missile test that he was "very disappointed" in China and that Beijing profits from U.S. trade but had done "nothing" for the United States with regards to North Korea, something he would not allow to continue.
Asked by a reporter on Monday how he plans to deal with Pyongyang, Trump said at the start of a Cabinet meeting: "We'll handle North Korea... It will be handled."
China's Foreign Ministry, in a statement sent to Reuters responding to Trump's earlier tweets, said the North Korean nuclear issue did not arise because of China and that everyone needed to work together to seek a resolution.
Russia said on Monday the United States and other countries were trying "to shift responsibility for the situation to Russia and China" following the most recent missile test.
"We view as groundless attempts undertaken by the U.S. and a number of other countries to shift responsibility to Russia and China, almost blaming Moscow and Beijing for indulging the missile and nuclear ambitions of the DPRK (North Korea)," the Russian Foreign Ministry said in a statement.
At the United Nations in New York, China's U.N. ambassador said on Monday it is primarily up to the United States and North Korea, not Beijing, to reduce tensions and work toward resuming talks to end Pyongyang's nuclear weapon and missile programs.
The United States and North Korea "hold the primary responsibility to keep things moving, to start moving in the right direction, not China," China's U.N. Ambassador Liu Jieyi told a news conference to mark the end of Beijing's presidency of the U.N. Security Council in July.
"No matter how capable China is, China's efforts will not yield practical results because it depends on the two principal parties," Liu said.
Chinese Vice Commerce Minister Qian Keming told a news conference there was no link between the North Korea issue and China-U.S. trade.
"We think the North Korea nuclear issue and China-U.S. trade are issues that are in two completely different domains. They aren't related. They should not be discussed together," Qian said.
China, with which North Korea does most of its trade, has repeatedly said it strictly follows U.N. resolutions on North Korea and has denounced unilateral U.S. sanctions as unhelpful.
Nikki Haley, U.S. Ambassador to the United Nations, said in a statement China must decide if it is willing to back imposing stronger U.N. sanctions on North Korea over Friday night's long-range missile test, the North's second this month.
Any new U.N. Security Council resolution "that does not significantly increase the international pressure on North Korea is of no value", Haley said, adding that Japan and South Korea also needed to do more.
Abe told reporters after his conversation with Trump that repeated efforts by the international community to find a peaceful solution to the North Korean issue had yet to bear fruit in the face of Pyongyang's unilateral "escalation".
"International society, including Russia and China, need to take this seriously and increase pressure," Abe said. He added Japan and the United States would take steps towards concrete action but did not give details.
Abe and Trump did not discuss military action against North Korea, nor what would constitute the crossing of a "red line" by Pyongyang, Deputy Chief Cabinet spokesman Koichi Hagiuda told reporters.
"Pyongyang is determined to develop its nuclear and missile programme and does not care about military threats from the U.S. and South Korea," state-run Chinese tabloid the Global Times said on Monday.
"How could Chinese sanctions change the situation?" said the paper, which is published by the ruling Communist Party's official People's Daily.
China wants both balanced trade with the United States and lasting peace on the Korean peninsula, its official Xinhua news agency added in a commentary.
"However, to realise these goals, Beijing needs a more cooperative partner in the White House, not one who piles blame on China for the United States' failures," it added.
The United States flew two supersonic B-1B bombers over the Korean peninsula in a show of force on Sunday in response to the missile test and the July 3 launch of the "Hwasong-14" rocket, the Pentagon said. The bombers took off from a U.S. air base in Guam and were joined by Japanese and South Korean fighter jets during the exercise.
"North Korea remains the most urgent threat to regional stability," Pacific Air Forces commander General Terrence J. O'Shaughnessy said in a statement.
"If called upon, we are ready to respond with rapid, lethal, and overwhelming force at a time and place of our choosing."
(Additional reporting by Chang-ran Kim in Tokyo, Christine Kim in Seoul, Michelle Nichols and Riham Alkousaa at the United Nations, Polina Devitt in Moscow and Steve Holland in Washington; Editing by Michael Perry and James Dalgleish)
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