Technically, no. Gum does not break the Communion fast, and Catholics ordinarily have a right to receive the sacraments. The priest should have given your friend Communion and then later counseled her privately about the inappropriateness of chewing gum at Mass, much less approaching for Communion with gum in her mouth.
That said, the fact that your friend has suffered some mild embarrassment – whether or not the priest was correct in allowing it to happen – is not entirely a bad thing. Perhaps she will learn from the experience and think more deeply about consideration for her Lord and Savior Jesus Christ (who suffers when he is treated shabbily by apathetic communicants who do not think about how they are receiving him into themselves) and consideration for her fellow Christians (who suffer to see the Mass treated by their fellow congregants with less dignity than your local courtroom).
