Why did Jesus refer to Gentiles as dogs?

I disagree with all above interpretations.

Jesus was very wise and obviously familiar with the Greek culture which still extended into his territory of Nazareth and environs. In Greek culture, being a dog wasn't an insult, but rather what we label today as a skeptic. As such, Jesus is simply making the distinction in this passage between the Children of Israel and the Gentiles in the sense that the Children of Israel are chosen and their religion expects them to believe in the Jewish God;

However, the gentiles of which this woman was, are assumed not to believe in the Jewish God, as after all they were not Jewish. It is this reasoning Jesus follows, and the word dog might make more sense if you yused the modern synonym to the Greek Gospel for "Dog" and contemplated it as "skeptics".

In Jesus' wisdom he was impressed by the humility and thoughtfulness of this woman, who rather than affirm she was a skeptic, was very clever in flipping it right back to Jesus by herself making use of the double meaning and basically demonstrating her faith in him as Master by her tenacity and subsequent witty reply. She affirms she is a believer, and even if he wishes to call her a skeptic, her words say otherwise as she puts her faith in Him and lets Him call her however he wishes without complaint.

This is typical of Jesus' own path and those who follow.

Firstly, if it is of any consolation, he did not call her dog in an insulting sense as kuōn G2965 (Matt 7:6) "do not give what is holy to dogs"; but the word used here is kunarion G2952 can be translated as pet dogs or little dogs.

The the reason why Jesus and the apostles uses such language is due to the particular cultural norm; It may appear racist insult to someone in today's time but it was the general understanding of Jews concerning pagans. The Canaanite woman was a pagan as opposed to Samaritan woman who was a half-breed. The incidents show that Jesus was kind and compassionate towards pagans and women compared with other religious Jews.

On John 4:7 Charles Ellicot comments

Josephus spoke of this as the customary way of the Galileans going up during the feasts at Jerusalem (Ant. xx. 6, § 1). The Pharisees, indeed, took the longer road through Peræa, to avoid contact with the country and people of Samaria, but it is within the purpose of His life and work (“needs go,” i.e., was necessary that He should go) to teach in Samaria, as in Judæa, the principles of true religion and worship, which would cut away the foundations of all local jealousies and feuds, and establish for all nations the spiritual service of the universal Father (Joh. 4:21-24).

Jesus had just finished using sheep as an illustration when talking with his Jewish disciples - an illustration that they would understand. I doubt Jewish people of that day typically kept dogs as pets. Yet when talking with a woman whose culture evidently did keep dogs as pets, he uses it as an illustration, not to drive her away, but to communicate effectively with her. His comments leave her thinking of all the times she had told her young daughter, now suffering with no other remedy in sight but Christ's healing, not to sneak food off the table for her puppy. The woman seemed to understand that he, as the child of the Heavenly Father, was not to change his focus from accomplishing what he came to do, in order to dedicate his time to reach Gentile populations. Yet she reminded him that his work would inevitably have a spillover effect for good on all gentiles around (praise God for that!). She seemed in no wise belittled by his illustration using pets, nor am I.

It was an insult, but also a challenge. As one of the Fathers said, if you wish to test a man for pride, see if he can bear insult. It is important to consider that to the Jews of that time, Gentiles were 'dogs': they did not keep the law or even have the law, so they, like dogs, not only lived uncleanly but did not have the understanding (because they lacked the law) to even act in accordance with God's will, and thus were 'bestial'.

It demonstrated that true faith, as this woman had, was not barred by racism, phyletism, prejudice, class division, or any other thing. In his ironic use (given that he knew, as God, that she would overcome the insult) of this slur, he demonstrates that 'in him there is no Jew or Gentile, but all are one.'

My wife added this:

It was a compliment, because God knew beforehand she could bear the insult. If she would have been 'offended' and turn away, he would never have used the slur.