Russia must end its "irresponsible and destabilising activity", Theresa May has told President Vladimir Putin in one-to-one talks, Downing Street said.
Describing the Salisbury poisoning as a "truly despicable act", she said the two suspects - believed to be Russian military intelligence officers - must be "brought to justice".
Mrs May said the UK was open to a "different relationship" but that "this behaviour could never be repeated".
The Kremlin denies any involvement.
The UK believes two officers from Russia's military intelligence service, the GRU, were behind the poisoning of former spy Sergei Skripal and his daughter Yulia, in March 2018.
Conservative leadership candidate Boris Johnson has said Russia will "always let you down", adding it was "very difficult to find any defence for their behaviour" when it came to the Salisbury poisonings.
Speaking at the latest Tory party hustings in Exeter, he said there is a "global sense of repulsion at the way Russia behaves".
Mr Johnson added that "every British prime minister, every foreign secretary" in the past 10 years thought they could have a "normalisation" with Russia but were let down.
Meanwhile, Jeremy Hunt, running against Mr Johnson to be the next prime minister, has called comments made by Mr Putin about Western liberalism "an absolute disgrace".
"It's all very easy to be disparaging about democracy when you've never had to bother with it at home," Mr Hunt said.
Referring to the attack on Sergei and Yulia Skripal, he said: "My worry about Russia is that they're up to their old tricks."
'Fuss about spies'
Mrs May and Mr Putin met for talks on the margins of the G20 summit in Osaka, Japan, on Friday - the first formal bilateral meeting between the two leaders since the Salisbury incident.
It comes as Mr Putin, in an interview with the Financial Times, dismissed the Skripal poisoning as a "fuss about spies" and attacked liberalism, a core principle of western democracy.
As they met, the two leaders shook hands but there appeared to be little warmth, with Mrs May looking serious and unsmiling, as the pair began their meeting.
Downing Street said Mrs May had told the Russian president that the use of a deadly nerve agent on the streets of Salisbury formed part of a wider pattern of unacceptable behaviour.
She said the UK had irrefutable evidence that Russia was behind the attack that led to the death of a British citizen, Dawn Sturgess.
Mrs May also said the UK would continue to "unequivocally defend liberal democracy" and protect the human rights and equality of all groups, including LGBT people.
After the issue of Mrs May's facial expression was raised by a reporter on Russian TV, Vladimir Putin's press secretary, Dmitry Peskov, praised the prime minister for her "friendly and constructive attitude".
Mr Peskov said there had been a "a long pause in communication" between Russia and Britain, adding that "the fact itself that the meeting took place is very positive."
Speaking to BBC deputy political editor John Pienaar ahead of the meeting on Friday, Mrs May said the UK had already set out evidence on the attack in Salisbury last year and made charges against two Russians.
"Russia does not allow the extradition of its nationals, but European arrest warrants are out for those two individuals and if they set foot outside Russia we will be making every effort that they are brought to justice," she said.
"Russia needs to recognise its acts and stop acting in this way - and stop its other destabilising activities around the world, including, for example, its use of disinformation and cyber-attacks," she added.
Mr Skripal and his daughter both survived the poisoning in March last year, but a British woman, Dawn Sturgess, died last July after coming into contact with the nerve agent Novichok through handling a contaminated perfume bottle.
Scotland Yard and the CPS say there is sufficient evidence to charge two Russians - who go by the names of Alexander Petrov and Ruslan Boshirov - for attempted murder.
However, Mr Putin has insisted they are civilians, not criminals.
He said he hoped "a few preliminary steps" would be made towards restoring relations between Russia and the UK.
'Must be punished'
However, he appeared to have little sympathy for Mr Skripal. The former Russian military intelligence colonel was jailed after being accused of spying for Britain, before being released as part of a spy swap.
"Treason is the gravest crime possible and traitors must be punished," Mr Putin told the newspaper.
"I am not saying that the Salisbury incident is the way to do it. Not at all. But traitors must be punished.
"Listen, all this fuss about spies and counterspies, it is not worth serious interstate relations. This spy story, as we say, it is not worth five kopecks. Or even five pounds, for that matter."
The Russian President has previously labelled Mr Skripal as a "traitor" and a "scumbag".