In a post-race interview, Balding urged the 23-year-old to show his teeth and told him he could afford to "get them done" with his prize money.
A BBC spokeswoman said the presenter "had no intention whatsoever" of upsetting or embarrassing the jockey.
The BBC received 1,477 complaints from viewers over the remarks.
Treadwell won Saturday's race on 100-1 outsider Mon Mome.
In fairness they were pretty minging.