Foreign Secretary Boris Johnson has reassured Russians that the UK's dispute is specifically with "Putin's Kremlin", claiming it is "overwhelming likely it was his decision to direct the use of a nerve agent" in Britain.
The escalating global scandal is unfolding as Skripal and his daughter remain in critical condition after exposure to the Soviet-designed chemical Novichok on March 4 in the English city of Salisbury.
Speaking on a visit to the Battle of Britain Bunker museum in Uxbridge with his Polish counterpart Jacek Czaputowicz, Mr Johnson said: "Our quarrel is with Putin's Kremlin, and with his decision - and we think it overwhelmingly likely that it was his decision - to direct the use of a nerve agent on the streets of the United Kingdom, on the streets of Europe, for the first time since the Second World War".
"I'm absolutely certain that Russian Federation has underestimated the resolve and unity of North Atlantic Treaty Organisation allies when we have implemented different kinds of sanctions over the last years", Mr Stoltenberg added.
Russia has denied any involvement, cast Britain as a post-colonial power unsettled by Brexit, and even suggested London fabricated the attack in an attempt to whip up anti-Russian hysteria.
The Kremlin's Peskov called the allegation that Putin was involved "a shocking and unforgivable breach of the diplomatic rules of decent behaviour", TASS news agency reported.
He said the "UK is not alone" and North Atlantic Treaty Organisation allies give "strong political support" to Britain, following a joint statement from the US, France and Germany backing Mrs May's government and a pledge of support from Australia. Britain blamed Russian Federation for the attack. He also stated that while Russian Federation had nothing to gain from such an attack, "I can think of a great number of countries that would benefit from such accusations".More news: Petr Cech delighted to end 'frustrating' wait for landmark clean sheet
He said Russian Federation would target the same number of people as the USA but didn't say what the sanctions would involve.
"We demand material proof of the alleged Russian nerve agent traces from this event".
The source of the nerve agent used - which Britain says is Soviet-made Novichok - is unclear.
Britain has blamed Russia for the poisoning and announced the expulsion of 23 Russian diplomats.
Lavrov said Russian Federation will "of course" expel British diplomats and that he hopes the Skripals recover soon so light can be shed on what happened.
Mr Lavrov said: "I guess he wants to go down in history with some bombastic statements ..." Labour Party leader Jeremy Corbyn said in a newspaper column that politicians must not "rush way ahead of the evidence being gathered by the police".
Meanwhile, British Defense Secretary Gavin Williamson said Moscow had made a "deliberate political decision" to poison Skripal. The United States, France and Germany also condemned Russian Federation over the attack.