If we say something highly offensive, digging in our heels is definitely disgraceful, but quietly taking down the post is not enough, especially if many have protested. There is healing in a frank admission of wrong, that does not minimize nor excuse our own failings. It can't change what we've done, nor does it require that trust be immediately restored, but it is a healing moment, leading towards reconciliation.
The power of an apology is that it validates the person we hurt.
It views them as a person created in the image of God. If the people of God are to be childlike in innocence and trust, I do think it is childish to refuse to apologize when we cause deep pain to anyone. The context of victims of sexual abuse and their families is even more so an issue. Perhaps those who refuse to apologize for their reckless words are afraid of lawsuits. But do they fear God?
God's love releases us from fear, and helps us to take responsibility for consequences, knowing we are secure in his love, no matter what. I don't understand "Christian" obfuscation in the light of the cross, in fact, such a thing is not at all "becoming" of a "Little Christ".
Humility is the way of the cross, and instead of hiding sins, a humble apology from those who refused to listen to victims, then even misrepresented them, would not go amiss. Maybe you could lose your ministry, but you will find mercy and be a source of true healing. That ministry is the power of an apology, and why it matters so much.