* Putin acknowledges some influence over rebels
* Russian leader hits back at Western critics
* Putin considers economic, military steps (Adds details of comments)
By Darya Korsunskaya
MOSCOW, July 22 (Reuters) - President Vladimir Putin said on Tuesday Russia would try to ensure Ukrainian separatists cooperate with an investigation into the downing of a Malaysian airliner, but said the West must do more to persuade Kiev to end hostilities.
Putin came out fighting in his most detailed comments since the plane was brought down on Thursday, dismissing criticism of Russia's role in events in rebel-held east Ukraine and describing the West's position as "strange and unacceptable".
Accusing the United States indirectly of pulling the strings in Kiev, trying to bully Russia and meddling in Russia's domestic affairs, the president said in televised remarks: "Such methods will not work on Russia."
Reading from notes at the head of a long table flanked by his top government, parliament, security and defence officials, Putin spoke much more forcefully than during brief televised remarks on the plane's downing first released in the early hours of Monday, when he looked tired and less assured than usual.
He did not, however, directly address accusations by the United States and Ukraine's pro-Western leaders that Moscow is supplying the rebels with arms, including the missile system used to bring down the airliner, and his promise to use Russia's influence with the separatists was vague.
"We are being called on to use our influence with the separatists in southeastern Ukraine. We of course will do everything in our power but that is not nearly enough," Putin told a meeting of his advisory Security Council.
It was a rare acknowledgement that Russia has influence over the rebels, echoing a similar remark by his foreign minister, but he made clear Washington should be doing more to use its sway over the authorities in Kiev, described by a top security aide as the West's henchmen.
"Ultimately, there is a need to call on the authorities in Kiev to respect basic norms of decency, and at least for a short time implement a ceasefire," he said, repeating criticism of Kiev for resuming military operations after a truce.
The former KGB spy also did not respond directly to calls for Russia to tighten controls at the border with Ukraine, which the West says would help prevent arms reaching the rebels who oppose Kiev's rule over the Russian-speaking east.
DETERMINED NOT TO LOOK WEAK
Putin had previously made only short comments in public on the downing of flight MH17, killing all 298 people on board - the televised remarks standing beside a desk early on Monday and comments filmed at the start of a meeting on Friday.
His new remarks appeared partly to respond to U.S. President Barack Obama, who urged him and Russia on Monday to "pivot away from the strategy that they've been taking and get serious about trying to resolve hostilities within Ukraine".
Obama and other Western leaders have painted this as a last chance for Putin to end the worst crisis in Moscow's relations with the West since the Cold war, hoping he will distance himself from the rebels and cut off any support for them.
Putin has an interest in de-escalating the crisis to avert more Western sanctions on Russia and reduce the risk of events spinning further out of control in east Ukraine. He signalled this by renouncing powers given to him by parliament to send Russia's army into east Ukraine to protect Russian-speakers.
But he is also determined not to be seen to make big concessions - something which would damage his popularity ratings in Russia, which have soared to record highs since the annexation of Crimea in March.
Hitting back, but repeating earlier vague threats, he said Russia, which has been hit by Western sanctions over Crimea, could take steps to protect the economy from "external threats" and strengthen its defences to counter moves by NATO in eastern Europe.
Putin also reiterated his belief that protests that toppled Ukraine's former Russian-backed president Viktor Yanukovich in February were an illegal coup instigated and funded from abroad.
"Russia is being presented with what is almost an ultimatum: 'Let us destroy this part of the population that is ethnically and historically close to Russia and we will not impose sanctions against you'," Putin said.
"This is a strange and unacceptable logic."
Earlier on Tuesday, Putin signed into law tougher punishment for public calls for separatism in Russia, including up to four years in jail, but he denied any plans to clamp down on civil society. (Writing by Alissa de Carbonnel; Editing by Timothy Heritage)