A lawyer for Scotland Yard told the Leveson Inquiry last week it was "unlikely" News International journalists erased messages from the schoolgirl's phone three days after she went missing in 2002.
The Guardian published a correction to its original story from July 5 which first made the allegation.
This has also been added to the bottom of the online versions of about 20 later articles which repeated the claim.
Richard Caseby, managing editor of The Sun, wrote to the Guardian's editor, Alan Rusbridger, and its readers' editor, Chris Elliott, on Friday asking for corrections in print and on the paper's website of 26 articles published since the first story.
Caseby said: "Alan Rusbridger has corrected the gross inaccuracy of his original statement of fact that the News of the World deleted the voicemail of Milly Dowler and gave her parents false hope she was alive.
"However, that statement of fact was repeated in 26 subsequent Guardian articles and the record now needs to be corrected with respect to those, both in print and online.
"Alan Rusbridger has a destructive agenda against the entire popular press, as evidenced by the regularity by which he publishes false stories about the News of the World and The Sun.
"Since July I have demanded and received three apologies and corrections for extremely damaging false statements of fact about The Sun.
"Alan Rusbridger likes to think of himself as stepping down from Mount Olympus to lecture the rest of the newspaper industry on journalistic ethics. But he could make a start by cleaning up his own stable."
A Guardian spokeswoman said: "Mr Caseby is currently dealing with the Guardian's independent readers' editor over matters which he raised only on Friday.
"We have already promptly footnoted a number of articles and are considering a number of others.
"In addition, we have noted in print, including on the front page, the fact that the police now believe that is it 'unlikely' that voicemail deletions by News of the World journalists caused Milly Dowler's parents false hope.
"We note that, even this week, News International have refused to confirm or deny whether News of the World journalists did, in fact, delete Milly's messages. That remains an open question.
"That the News of the World repeatedly, and with senior executive approval, hacked a dead teenager's phone is not in dispute. At the time the News of the World was closed the company's chief executive warned of worse revelations to come which would eventually explain the company's decision to kill the title.
"Alan Rusbridger does not hold the views about popular journalism ascribed to him by Mr Caseby."
Lord Justice Leveson, who is chairing the public inquiry into press standards, has asked police to get to the bottom of how Milly's voicemails came to be deleted.
"My present view is that this has achieved such a significance that it cannot be left alone," he said at a hearing last week.