Clarke was fined 20 percent of his match fee on Monday after being caught by a stump microphone telling the fast bowler to expect a broken arm in the final stages of Australia's 381-run victory in Brisbane Sunday.
The usually mild-mannered Clarke also wagged his finger in a close-up confrontation with the England quick, a renowned sledger, with the umpires needing to step in to cool tempers.
As a member of the Nine Network's commentary team, Warne had access to all the stump microphone recordings and tweeted that Anderson had threatened Bailey, who was fielding in close, with a punch to the face.
"I think it's a disgrace that @MClarke23 has been fined. What about what Jimmy Anderson said to Bailey, which wasn't heard," Warne posted on Twitter late Monday.
"@MClarke23 stuck up for his debutant Bailey as he should have too as capt after Anderson said he wanted to punch Bailey in the face.
"Unfortunately only Clarke's reaction to Anderson's was heard live, we all heard Anderson's sledge that led to Clarke reacting."
He added: "Lastly as an international sportsman, if you decide to dish it out ie sledging you have to be prepared to cop it back too & not complain."
Bailey has remained tight-lipped on the Anderson incident, saying only that "he must have been just a bit upset about the way the game was going I guess".
Channel Nine on Tuesday apologised that the Clarke comment was broadcast. Under an agreement between Cricket Australia and the television network, the stump mike is not supposed to be on between balls so his threat should not have been recorded.
"He is an outstanding Australian cricket captain who's just led his team to a marvellous and historic win," Nine's Head of Sport Steve Crawley told reporters.
"He doesn't need or deserve that to be tarnished by our error and we of course apologise."
The explosive incident capped a torrid first Ashes Test with England skipper Alastair Cook also fuming at David Warner calling his team frightened, while singling out batsman Jonathan Trott as "weak".
Matters took a dramatic turn late Monday when Trott, a stalwart of the team, pulled out of the Ashes series and returned to England due to a "stress-related illness", widely reported to be depression.
Prior to the news of Trott's departure, Warner admitted that he "probably went a little bit too far", although England coach Andy Flower said the batsman's departure was not directly related to the comments.