'Speaking in a tense meeting at the G20 summit in Japan, the Prime Minister warned Mr Putin that Russia must adopt a "different path" if it wanted a "different relationship" with the UK.
She also insisted that the two suspects in last year's chemical weapons attack in Salisbury are "brought to justice", after Russia refused to extradite the pair.
The two leaders were speaking in their first formal meeting since nerve agent attack, for which Mrs May has said it was highly likely the Russian state was responsible.
In a riposte to Mr Putin's earlier claim that liberal values are obsolete, Downing Street said that Mrs May told the Russian president that Britain would "continue to unequivocally defend liberal democracy" as well as "protect the human rights and equality of all groups, including LGBT people."
A Downing Street spokesman said: "[The Prime Minister] told the President that there cannot be a normalisation of our bilateral relationship until Russia stops the irresponsible and destabilising activity that threatens the UK and its allies - including hostile interventions in other countries, disinformation and cyber attacks – which undermine Russia’s standing in the world."
The talks came after Mr Putin used an interview with the Financial Times to claim that Angela Merkel, the German Chancellor, had made a cardinal mistake by adopting a liberal policy towards immigration from the Middle East.
The claim prompted a sharp response from Jeremy Hunt, the Foreign Secretary, who said Mr Putin was attempting to "subvert our values from abroad."
"Our democratic system based on rules, respect for individual rights & human dignity has created more happiness & wealth than any other way of life anywhere in the world," he said.'
Read more: Theresa May calls on Vladimir Putin to halt 'irresponsible' activity around the world
Did you like this article?
Thank you for your vote!
29 June 2019
'6 or 5 genders… no idea what that even is': Putin says LGBT should not enforce their agenda on kids
From our advertisers
Epstein: Gateway to the Global Cesspit - by David Icke who has been exposing elite paedophiles and Satanists for decades
Why they say Epstein is dead - Epstein 'survivor' talks about him and the paedophile/Satanic elite
From our advertisers