During prime minister's questions last week, Gordon Brown said: "This is a very serious matter indeed and I hope it is investigated by the authorities."
However, the Electoral Commission said it would not launch an inquiry, as "soliciting a donation is not an offence".
Mr Osborne told The World at One: "I think the real judgment is can you learn from mistakes you make. I have changed the way that I am going to operate when it comes to fundraising and I will not discuss individual donations with individual donors.
"That, I think, is an appropriate thing for me to do."
David Cameron said his shadow chancellor had made it "clear" that he would not be involved in the details of party fundraising any more.
Pressed about what lessons the Conservatives had learnt from the donation row, Mr Cameron told the BBC: "We have all learnt lessons from this. There is no doubt about that. George Osborne has been very frank about explaining what they are."
For Labour, Transport Secretary Geoff Hoon said: "Now that George Osborne has admitted that he made a mistake in Corfu, when will he admit the rest of the mistakes he has made, like pledging to hike up fuel tax at the worst time for families and business?"
He added: "George Osborne's judgement is wrong. And, once again, it is because he creates headlines on the hoof without care for the consequences. His mistakes just go to show why this is no time for a novice."
Liberal Democrat Treasury spokesman Vince Cable said: "He [Mr Osborne] did show very bad judgement in meeting this guy [Mr Deripaska] five times.
"If he is effectively apologising I am sure that is the line under his particular involvement."
David Cameron: 'We have all learnt lessons from this'
Controversy has surrounded a party on board Mr Deripaska's yacht, which was attended by Business Secretary Lord Mandelson when he was still the European Trade Commissioner.
Last week, the minister admitted first meeting the aluminium magnate in 2004 - two years earlier than officials had previously said.
Lord Mandelson's relationship with Mr Deripaska has led some to suggest he could have intervened in EU policy on the Russian's behalf.
In 2004, Lord Mandelson proposed that tariffs on aluminium imports be cut from 6% to 3%.
But on Sunday he told Sky News: "A lot has been said about the relationship I have with one particular Russian businessman.
"All I would say about that is that he has never asked for any favours, I have never given him any favours and that is what the European Commission in their examination of the issue has very firmly put on record."
On Sunday, the Conservatives urged Lord Mandelson to "get things straight" over his dealings with Mr Deripaska.
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