A military-grade nerve agent - a weapon of mass destruction - has been used in an attempt to kill civilians on British soil in a reckless fashion, she said.
Yulia is conscious and talking in hospital.
Britain blamed Russia for the poisoning, calling it attempted murder, and asked Moscow to explain what happened but Russia denies any involvement and has suggested Britain itself carried out the attack to stoke anti-Russian hysteria.
Yulia, a Russian citizen, had arrived in Britain only the day before to visit her father, who has been living in Britain for some seven years. Pierce said: "There is one country, Russia, which is playing a fast and loose with our collective security and the worldwide institutions that protect us".
"I made this recording, that was Yulia", she said.
The first public comments by Yulia Skripal since the poisoning emerged on Thursday. However, senior United Kingdom officials told The Times that father and daughter "would likely never be the same again" and could "require ongoing medical care".
As Yulia herself says, her strength is growing daily and she can look forward to the day when she is well enough to leave the hospital.
"Any speculation on when that date will be is just that - speculation".
Her comments were dismissed by a former British ambassador to Russian Federation as part of an "absurdist" and "cunning" attempt by the Kremlin to influence public perception of the ongoing dispute between the two countries.
He said: "They can speak to investigators about their last movements, were there any threats, were there any other circumstances?"
Writing on Twitter, UK Foreign Secretary Boris Johnson welcomed the news of Skripal's improved condition and wished both patients a "quick return to full health".
The British Foreign Office confirmed it had received the request.
"The NHS will continue to provide ongoing care for the Skripals, both of whom are likely to have ongoing medical needs", the FCO said.
"This botched double murder attempt was defeated by the brilliance of British scientists and the doctors in Salisbury who, under enormous pressure, came up with a bespoke set of treatments to thwart a boutique chemical weapon specifically designed for assassinations", he told the Mail on Sunday.
At the United Nations Security Council meeting on Thursday, Vassily Nebenzia, Russia's UN envoy, called the allegations of Russian culpability a "fake story" and said the UK was "playing with fire and they will be sorry".
Moscow has retaliated by expelling an equal number of Western diplomats, questioning how Britain knows that Russian Federation was behind the attack. She thanked those who had treated her and her father and all those who had sent "messages of goodwill".
She said: "I am sure you appreciate that the entire episode is somewhat disorientating, and I hope that you'll respect my privacy and that of my family during the period of my convalescence".
Russian Federation "is not responsible" for the attack in Salisbury, that country's Ambassador to the UN, Vassily Nebenzia, said, stressing that the origin of the substance used in the incident has not been confirmed.