The ad showed an adult leafing through the pages of a book entitled "Baby Nutrition".
Three chapters of the book showed a baby growing from six-months-old into a child about to walk towards a women who lifts him into her lap.
A voiceover stated: "As it grows, a baby needs a special combination of nutrients to sustain the incredible growth in its brain, body and immune system.
"To provide for those three essential aspects of growth, Heinz created new Nurture, an advanced complete follow-on formula to help nourish, protect and develop your baby."
Three viewers, which was created by Abbott Mead Vickers BBDO and ran during July 2008, claimed it "implied health claims with regard to children's immunity and development which they believed could not be substantiated".
After a what it called a "complex" investigation the ASA agreed, ruling that the "evidence submitted [by Heinz] was not sufficiently robust to support the products claims in relation to children's immunity and development".
An ASA spokesman said: "The adjudication involving Heinz took 18 months because of the complexity of the issues being investigated. There was some significant EC legislation to examine and a swathe of evidence and robust counter argument put forward by the advertiser in its defence of its ad that we are duty bound to assess. We have to give an advertiser every opportunity to defend itself."