The Catholic Church, of course, cannot trademark or copyright Catholicism or its doctrine, and so in most locales she has no secular legal standing to restrict the use of the word or term "Catholic".

However, the bishop can and will have a say in which Catholic groups call themselves "Catholic" and represent themselves as Catholic groups within the Catholic Church, and so they have ecclesiastical authority to revoke that right of representation. Not a few groups have abused the honor and privilege of the Catholic name, and it's been revoked.

Now also, whether or not a group chooses to cease calling itself "Catholic" is up to the group entirely, and whether they choose to be obedient. The National Catholic Register notoriously brushed off their bishop's concerns and continues to use the name. Others simply rename to something that doesn't contain that word.

It's also the case that the bishop gets to say who can reserve the Blessed Sacrament, and have it in a chapel where Mass is said. For example, St. Joseph Hospital and Medical Center in Phoenix had those privileges revoked, along with their Catholic identity, when they performed an abortion. That identity was later restored by the same bishop who revoked it, so it's all good.