The billboard advert shows a toddler smoking a cigar with the tagline: "You wouldn't let your child smoke. Like smoking, eating meat increases the risk of heart disease and cancer. Go vegan! Peta."
The ASA told the charity last week that the advert must not appear again in that form after ruling that it breached rules on misleading advertising and substantiation.
The watchdog received two complaints about the advert and asked Peta to back up its claims.
Peta told the ASA the link between meat and an increased risk of heart disease and some cancers was well documented and submitted a number of medical studies to support its claim.
But the ASA said the studies did not unequivocally support the claim and that it should not imply that any consumption of meat would increase the risk of heart disease and cancer.
Peta said today that it would refer the ASA’s ruling to the independent reviewer and told Third Sector it had submitted more than 30 new pieces of medical research in support of its claim.
Mimi Bekhechi, associate director of Peta, said: "We don’t agree with the ASA’s rulings on misleading advertising and substantiation because of all the evidence that is available. We hope the appeal is successful."
The ASA said appeals must establish that a substantial flaw of process or adjudication is apparent or show that additional relevant evidence is available.
The independent reviewer can decide not to proceed with an appeal if it does not meet these criteria.
Of the 57 non-broadcast appeals to the independent reviewer in 2012, 29 did not meet the required criteria and two resulted in rulings being reversed.