Downing Street said ministers will not attempt to "micro-manage" the largely state-owned bank. It was for the board to set bonuses and for individuals to decide whether to accept them, said a spokeswoman.
But Labour leader Ed Miliband insisted that Mr Hester's decision to waive an award worth almost £1 million did not draw a line under the issue of City pay.
Labour will use an opposition day debate in the Commons next Tuesday to urge Conservatives and Liberal Democrats to back a £2 billion tax on bankers' bonuses, which they say could fund the creation of 100,000 jobs elsewhere.
A senior party source said: "This debate is about more than Stephen Hester. It is time for the Conservatives and Lib Dems to show they are serious about tackling the bonus culture by supporting our plan to repeat the bank bonus tax.
"The banks, who helped create the financial crisis, must now help return our economy to growth. We need a bonus tax, we need to give employees a say on top pay in every company, and we need to change the rules and culture so that bonuses are a reward for exceptional performance rather than what you get for doing your job."
Labour had initially planned to use next week's debate to force a vote calling for Mr Hester to be stripped of his bonus, and Mr Miliband said that it was this threat which persuaded the RBS boss to give up his £963,000 shares package last night.
The Labour leader said: "The Government has dragged its feet, has nodded this bonus through, and I think we have seen that they have got a completely tin ear when it comes to understanding what people are feeling.
"I don't think this can be just a one-off episode, because if we don't deal with this systematically, if we don't deal with the issue of bankers' bonuses in a proper way, this kind of thing is just going to re-occur."
Ministers including Chancellor George Osborne today welcomed Mr Hester's announcement.
"This is a sensible and welcome decision that enables Stephen Hester to focus on the very important job he has got to do, namely to get back billions of pounds of taxpayers' money that was put into RBS," said Mr Osborne.
And Business Secretary Vince Cable said: "This is a welcome announcement and a positive step towards curbing the bonus culture.
"RBS should now focus on what the bank should be doing, which is lending to good British companies."
But Downing Street insisted that ministers would not be "micro-managing" RBS and that decisions on future bonuses at the 83% state-owned bank were a matter for the board.
A number of executives in RBS's investment branch are expected to be offered large payments in the coming weeks, with a bonus pool expected to be around half of last year's £950 million.
A Number 10 spokeswoman said: "These decisions are decisions for the board, in terms of who gets a bonus and what they get. The Prime Minister has made very clear that bonuses should be responsible."
It was "a matter for individuals" whether they choose to accept any bonus they are offered, she said, adding: "Reward for good performance is not in itself a bad thing."
Mr Hester was appointed chief executive at the end of 2008 to replace Sir Fred Goodwin, after the bank had to be bailed out by the government.
His contract stipulated that he should be considered for a bonus based on performance against a set of goals including increasing lending and reducing the bank's overblown £2 trillion balance sheet. Although RBS's share price has tumbled by more than a third over the past year, Number 10 today said he was making "good progress" on these targets.
But Unite union national officer David Fleming said it was "common sense" for bosses to waive bonuses at a time when RBS was imposing large numbers of redundancies among frontline staff.
"This gesture goes some way in acknowledging the hypocrisy of an organisation which has sacked over 21,000 staff, while still attempting to pay bumper bonuses to the bosses," said Mr Fleming.
"There remains a long way for RBS to go in proving its credentials as a responsible organisation, to its customers and also to its thousands of staff."
David Hillman, of the Robin Hood Tax campaign for a financial transactions tax, said: "We're delighted that public pressure has trumped Government inaction and has persuaded Mr Hester to give up his bonus, but RBS is just the tip of the iceberg and we cannot rely on the consciences of bankers to relinquish their excessive pay.
"If the Government is serious about reining in the banks, it must go beyond rhetoric and take concrete action to ensure the financial sector pays its fair share to society."
PA
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