Mark Thompson also addressed criticism of the BBC from News Corporation chief James Murdoch, who last month described the scale and scope of the BBC's ambitions as "chilling".
The director-general said Mr Murdoch existed in a "bi-polar" universe of market and state.
Mr Thompson took the platform at the Royal Television Society's (RTS) Cambridge Convention for a session entitled BBC Under Fire.
His speech followed calls from Mr Bradshaw for the BBC to stop expanding.
Mr Thompson said the BBC would "fight tooth and nail to preserve our broad public remit".
He found plenty of what Mr Bradshaw said to be "frankly puzzling".
Mr Thompson said most of the BBC's new services had not been approved by the corporation's governing body, the BBC Trust, but, with one or two exceptions, "they were approved by the Government of which Ben is a member".
"Indeed, the Government asked the BBC to launch a range of new services to help with their policy of encouraging the public to move to digital television and radio.
"Ben's surprise at these services is itself surprising."
Hitting back at Mr Bradshaw's questioning of the Trust as both "cheerleader" and regulator, Mr Thompson said: "The people Ben should ask this question of is those colleagues of his in the present Cabinet who invented the BBC Trust, approved it and enshrined it in a Charter which still has well over seven more years to run."
Mr Thompson admitted the corporation has "faults and failings, and of course there are plenty of them".
But he said: "At a time when the future of so much of the rest of media is so uncertain, the idea of the BBC still works.
"It works in terms of investment in production, in training, in talent.
"It works in innovation. But above all, it works for the public."
Mr Thompson said there was "no evidence" that a powerful BBC was making commercial broadcasters' problems worse.
"But we have to accept that to many in commercial media we seem relatively bigger and stronger than ever."
He said it would be "fatal" for the corporation to sit on its laurels.
He continued: "The public will be best served not by a strong BBC sitting in isolation but by a strong, varied media sector which includes a strong BBC. We get it."
Mr Murdoch, who is chairman and chief executive, Europe and Asia, News Corporation and non-executive chairman of BSkyB, highlighted the BBC as a "threat" to independent journalism in a speech at the Media Guardian Edinburgh International Television Festival last month.
Mr Thompson compared the BBC and other public service broadcasters with great British institutions such as universities, museums and galleries, occupying a "public space" rather than "James's bi-polar universe of market and state".
He said: "Public space is not-for-profit space, not by accident but by design.
"It exists not to make money but to serve the public.
"Ben Bradshaw was right last night to compare the way the public think of this kind of service with the way they think of the NHS."
Contrasting arts on the BBC with Sky Arts, Mr Thompson said: "The BBC exists in part to make the arts universally available, Sky does not.
"Private space focuses on the minority who already have a taste for the arts, public space reaches out across the population."
He said there was no exclusion with the BBC, adding: "You can't buy a better service from the BBC no matter how wealthy you are.
"And you can't stop people who are less well-off than you enjoying just as good a service as you do."
Last month, plans were announced to charge online customers within a year to view content across all News Corporation websites.
Mr Thompson said: "We will never erect a pay zone around our news.
"That's why we will fight tooth and nail to preserve our broad public remit - from Strictly to the Poetry Season."
Both Mr Thompson and BBC Trust chairman Sir Michael Lyons were present yesterday when Mr Bradshaw delivered his speech, and the politician later became involved in a heated exchange with Sir Michael.
Mr Bradshaw said: "I'm concerned about the regulatory structure of the BBC.
"Although the Trust has performed better than its predecessor, I don't think it is a sustainable model in the long term.
"I know of no other area of public life where - as is the case with the Trust - the same body is both regulator and cheerleader."
Mr Thompson said that, this autumn, the BBC would look ahead to the post-digital switchover world of 2012 and beyond to develop a clear strategy for how to best serve the public and support the media sector.
He promised: "The review will be both radical and open-minded. Ben Bradshaw wondered aloud last night whether the BBC might have reached the limits of expansion.
"Don't assume that we'll dismiss that notion out of hand or erect defensive barriers against it."
Mr Thompson said he was in agreement on many issues with Mr Bradshaw, including the "pride in the public service journalism" which the former BBC reporter turned politician felt.
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