Read Peter in context. He is referring to a Christian (not an animal) who turns away again to the world.
He's telling us why it is these people are that way.
2 Peter 2:9 "The Lord knoweth how to deliver the godly out of temptations, and to reserve the unjust unto the day of judgment to be punished"
If these dogs who return to their vomit are "the godly", then the Lord isn't very effective at delivering them from temptations.
In a parallelism, "the godly" and "the unjust" are contrasted, making them speaking of the same thing, one in the positive, and one in the negative.
Who are these "unjust" who are being reserved for judgment?
2 Peter 2:10 "But chiefly them that walk after the flesh in the lust of uncleanness, and despise government. Presumptuous [are they], selfwilled, they are not afraid to speak evil of dignities."
Them that walk after the flesh. Does this mean children of God who walk after the flesh?
Romans 8
1 There is therefore now no condemnation to them which are in Christ Jesus,
who walk not after the flesh, but after the Spirit.
2 For the law of the Spirit of life in Christ Jesus hath made me free from the law of sin and death.
3 For what the law could not do, in that it was weak through the flesh, God sending his own Son in the likeness of sinful flesh, and for sin, condemned sin in the flesh:
4
That the righteousness of the law might be fulfilled in us, who walk not after the flesh, but after the Spirit.
5 For
they that are after the flesh do mind the things of the flesh; but they that are after the Spirit the things of the Spirit.
6 For to be carnally minded is death; but to be spiritually minded is life and peace.
7 Because the carnal mind is enmity against God: for it is not subject to the law of God, neither indeed can be.
8 So then
they that are in the flesh cannot please God.
9 But
ye are not in the flesh, but in the Spirit, if so be that the Spirit of God dwell in you. Now if any man have not the Spirit of Christ, he is none of his.
10
And if Christ be in you, the body is dead because of sin; but the Spirit is life because of righteousness.
11 But if the Spirit of him that raised up Jesus from the dead dwell in you, he that raised up Christ from the dead shall also quicken your mortal bodies by his Spirit that dwelleth in you.
These "after the flesh" are not God's children. The contrasts in this chapter show this. There are those who are after the flesh, and those who are after the Spirit. You are not in the flesh, but in the Spirit, IF SO BE that the Spirit of God dwell in you.
2 Peter 2:10 "But chiefly them that walk after the flesh in the lust of uncleanness, and despise government. Presumptuous [are they], selfwilled, they are not afraid to speak evil of dignities."
They are called "dogs returning to their vomit" because they never were otherwise. That's what they are, so that's what they do.
Much love!