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Canon 915 states:
Those who have been excommunicated or interdicted after the imposition or declaration of the penalty and others obstinately persevering in manifest grave sin are not to be admitted to holy communion.
Those who publicly promote contraception are "persevering in manifest grave sin". It's manifest because it's public; it's grave because sexual sins are always grave matter.
Could public promotion of something that is against the teaching of the Catholic Church be used to excommunicate?
Yes
If yes, why doesn't the Catholic Church excommunicate her?
Cowardly bishops
Or who/where can claim the case for excommunication?
Bishops can excommunicate.