Livingstone made the comments about the newly appointed Wadley on Wednesday afternoon when he addressed the Greater London Assembly to explain his version of the events that have led to the row between the two.
He said: "I won't believe what I read in the Standard and I simply warn Londoners now: Don't believe anything you read about my policies or my private life in this paper until it's got a new editor."
The assembly was called yesterday afternoon to quiz the mayor after allegations were made in the Standard about his conduct at a 40th birthday party for his girlfriend Emma Beal's sister Kate.
The Standard reported that Livingstone had got drunk at the party and manhandled Emma Beal. When Beal's friend Robin Hedges, an art director at the Evening Standard's ES magazine, intervened, the paper claims Livingstone retaliated by pushing him over a 15ft wall.
Livingstone denies the allegations, saying that he and the pregnant Beal had a row after someone said she had smoked a cigarette, which turned out to be untrue.
Hedges is then alleged to have come running over to Livingstone and Beal, after someone told him that the row had turned violent, knocking the mayor to the ground.
Livingstone also says that he was nowhere nearby when Hedges, who the mayor describes as Beal's oldest and closest friend, fell over the wall, which he says is only 10ft.
In the statement, Livingstone said he would not be suing the Evening Standard for libel nor reporting it to the Press Complaints Commission because of bad experiences in the past.
"The truth is editors have more power than any cabinet minister. They have the power to besmirch a reputation and end a career. Now that power should only be used with the greatest of restraint," he said.
Livingstone added: "I think Londoners and the assembly have the right to say 'was that the case' in this instance. I believe Veronica Wadley was reckless with the facts and malicious with that power. For five weeks, we have had the Evening Standard trying to bring my mayoralty to an end, to ruin my reputation and hopefully, from their viewpoint, force me into resignation.
"Fortunately for me, Londoners decide who is to be mayor not editors, otherwise I wouldn't have been in this position in the first place."
Livingstone also said that he believes that if Wadley's predecessor Max Hastings was still at the helm, this would not have happened.
He said: "In the 21 years that I have been in the focus of public attention and unrelenting media attention, there has never been a single story linking me to violence or linking me to drunkenness.
"Perhaps if Max Hastings had still been editor he may have paused and considered whether there shouldn't have been some more detailed investigation before they ran with this story.
"Having run with this story they have consistently tried to justify it."
During the mayoral campaign in 2000, the Standard had backed Tory candidate Steven Norris over Livingstone.
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