Not at all. Let’s look at the passage in question.
But after hearing of Him, a woman whose little daughter had an unclean spirit immediately came and fell at His feet. Now the woman was a Gentile, of the Syrophoenician race. And she kept asking Him to cast the demon out of her daughter. And He was saying to her, “Let the children be satisfied first, for it is not good to take the children’s bread and throw it to the dogs.” But she answered and said to Him, “Yes, Lord, but even the dogs under the table feed on the children’s crumbs.” And He said to her, “Because of this answer go; the demon has gone out of your daughter.”
Jewish men of that time, by and large, did not speak to Gentiles (non-Jews). They especially did not speak to women. In fact, it’s said that they went so far out of their way to avoid contact with women and Gentiles that they would cross the street to avoid passing next to them.
Prior to this passage, Jesus miraculously fed 5,000 Jews in Jewish territory in chapter 6. Next, the beginning of chapter 7 shows Jesus giving the longest discourse in the NT on the interpretation of Jewish dietary laws. Jesus taught that all foods are clean…that what’s in your heart is more important than what’s in your stomach. Jesus proceeded to travel into Gentile territory and ate at a presumably Gentile home. On His way out, He had a conversation with a Gentile woman…evidence that He was NOT acting in concert with the racist and sexist views of many of His people.
The woman asked Him to heal her daughter. At first Jesus did not reply…we know this from the parallel passage in Matthew. She persisted, and Jesus spoke to her in a way that seems rude: He said that it’s not good to give the children’s food to dogs. Looking carefully tells us more, though: while the Jews used the word KUON for “dog”, Jesus didn’t use that word…He said KUNARION which means “small dog”, or more literally “puppy”. The woman surely saw the difference between Jesus’ response and the traditional Jewish response, and acted in concert, saying “even dogs get the children’s crumbs”. Jesus’ actions were consistent with the context: He demonstrated that the Jewish traditions regarding unclean foods and unclean people should be put aside. He spoke with a Gentile woman, healed her daughter, and proceeded (in the next chapter) to feed 4,000 Gentiles in Gentile territory.
The context of this story shows that Jesus’ words were not racist, but were instead a clear denouncement of racism, and of sexism as well.