The country's embassy in the UK fired off a salvo of seven tweets in which it said Britain must comply with the Chemical Weapons Convention for a joint investigation and warned the threat of sanctions would "meet with a response".
Meanwhile Russian foreign ministry spokeswoman Maria Zakharova told reporters: "One should not threaten a nuclear power".
On Monday, Vladimir Putin smirked when asked by BBC reporter Steve Rosenberg if Russia was behind the poisoning, saying: "First, get to the bottom of it [in Britain], then we'll discuss this."
In a series of tweets, the Russian Embassy said: "Moscow will not respond to London’s ultimatum until it receives samples of the chemical substance to which the UK investigators are referring.
"Britain must comply with the Chemical Weapons Convention which stipulates joint investigation into the incident, for which Moscow is ready.
"Without that, there can be no sense in any statements from London. The incident appears to be yet another crooked attempt by the UK authorities to discredit Russia.
"Any threat to take 'punitive' measures against Russia will meet with a response. The British side should be aware of that."
Mrs May said earlier this week the Government had concluded it is "highly likely" Russia was responsible for the attack which left ex-spy Mr Skripal and his daughter Yulia in a critical condition in hospital.
She demanded that Moscow account for how a Russian-produced nerve agent could have been deployed in Salisbury and vowed to set out measures Britain will adopt if no credible response is received by the end of Tuesday.
Home Secretary Amber Rudd earlier said Russia had "started responding" but the embassy appeared to suggest this amounted to little more than informing the Foreign Office of its demands and reiterating it was not involved.
Britain's ambassador to Russia was summoned by Moscow and told by first deputy foreign minister Vladimir Titov that the Kremlin "strongly protested" the accusations, the embassy said.
Russian Foreign Minister Sergey Lavrov said on Tuesday his country "is not to blame" for the poisoning.
Mr Lavrov said Russia had no connection to the poisoning, and demanded access to samples of the nerve agent used in the attack.
Mr Skripal, 66, was found along with his daughter, Yulia, on a bench in The Maltings shopping centre in Salisbury after police were called by a concerned member of the public at around 4.15pm on Sunday, March 4.
The pair did not have any visible injuries and were taken to Salisbury District Hospital where they remain in intensive care in a critical but stable condition. Detective Sergeant Nick Bailey, who went to the aid of Mr Skripal and his daughter, is still recovering in hospital.
Police officers in decontamination suits and military chemical weapons experts are continuing their investigations at several locations across Salisbury.
Mr Skripal and his daughter are both still fighting for their lives in hospital.
Additional reporting by the Press Association