This is one of those “garden variety” objections to Catholicism that come up so often. Like many such contra-Catholic arguments, it seems to have a plausibility at first glance. But upon closer examination, the alleged difficulty vanishes.
As is frequently the case, this “argument” hinges on an ignorance of different forms or genres of language in the Bible, and the importance of context. Critics of the Catholic Church frequently misunderstand both factors.
Matthew 23:9 (RSV) And call no man your father on earth, for you have one Father, who is in heaven.
That seems straightforward enough, doesn't it? How do Catholics get around this? Are we wantonly disregarding and disobeying a direct command of our Lord Jesus?
No, not at all. Jesus was simply using the common Hebrew teaching method of exaggeration or hyperbole. This is common in, for example, the book of Proverbs. Here are several other instances of the same sort of thing:
Matthew 7:3 Why do you see the speck that is in your brother's eye, but do not notice the log that is in your own eye?
Matthew 18:9 And if your eye causes you to sin, pluck it out and throw it away; it is better for you to enter life with one eye than with two eyes to be thrown into the hell of fire.
Matthew 19:24 Again I tell you, it is easier for a camel to go through the eye of a needle than for a rich man to enter the kingdom of God.
Matthew 23:24 You blind guides, straining out a gnat and swallowing a camel!
Luke 14:26 If any one comes to me and does not hate his own father and mother and wife and children and brothers and sisters, yes, and even his own life, he cannot be my disciple.
Jesus was making the point that God the Father is the ultimate source of all authority. He said this during the course of rebuking the Pharisees for spiritual pride (Mt 23:2-10). Those who try to reason in this way neglect to see that it would prohibit alluses of the word father whatsoever; even biologicalfathers. Since that is an absurd outcome, it is clear that the statement cannot be taken in an absolute sense.
Scott Hahn and Curtis Mitch, in The Ignatius Catholic Study Bible (San Francisco: Ignatius Press, 2010) comment on Matthew 23:9:
Jesus uses hyperbole to post a warning that no one should pridefully desire honorific titles. His words are not meant literally. . . . The spiritual fatherhood of New Covenant priests is an extension of its application to Old Covenant priests (Judg 17:10; 18:19).
Judges 17:10 reads: “And Micah said to him, ‘Stay with me, and be to me a father and a priest,. . .’” Judges 18:19 likewise uses the phraseology of “be to us a father and a priest.”
Jesus clearly didn’t have in mind a sweeping prohibition, since He Himself uses the term father many times (thus was not even following His own command, if the criticism is correct):
Matthew 15:4 For God commanded, 'Honor your father and your mother,' and, 'He who speaks evil of father or mother, let him surely die.' [from the Ten Commandments]
Matthew 19:5 . . . 'For this reason a man shall leave his father and mother and be joined to his wife, and the two shall become one flesh' . . .
Matthew 21:31 Which of the two did the will of his father? . . .
Luke 16:24 And he called out, 'Father Abraham, have mercy upon me, . . .' (cf. 16:27, 30)
John 8:56 Your father Abraham rejoiced that he was to see my day; he saw it and was glad.
St. Stephen, St. Paul, St. James, and the author of Hebrews use the term father, too:
Acts 7:2 And Stephen said: “Brethren and fathers, hear me. The God of glory appeared to our father Abraham, . . .”
Romans 4:12. . . the father of the circumcised . . . our father Abraham . . .
Romans 4:16-17. . . Abraham, for he is the father of us all, as it is written, “I have made you the father of many nations . . .”
Romans 9:10 And not only so, but also when Rebecca had conceived children by one man, our forefather Isaac,
1 Corinthians 4:15. . . I became your father in Christ Jesus through the gospel.
Philippians 2:22 But Timothy's worth you know, how as a son with a father he has served with me in the gospel. (cf. Philemon 10)
Hebrews 12:7, 9 . . . for what son is there whom his father does not discipline? . . .  Besides this, we have had earthly fathers to discipline us and we respected them.
James 2:21. . . Abraham our father . . .
Prophets were called father as well:
2 Kings 2:12 And Eli'sha saw it and he cried [to Elijah], “My father, my father! the chariots of Israel and its horsemen!” . . . (cf. 6:21)
Even Protestant leader John Calvin acknowledged that Jesus’ words couldn’t be taken literally, in his Commentaries, on Matthew 23:9:
[T]he honor of a father is falsely ascribed to men, when it obscures the glory of God. . . . strictly speaking, God alone is the Father of all.
It's a standard rule in hermeneutics (the interpretation of Scripture) that we interpret less clear passages of the Bible in light of clearer passages. This is a classic case. What seemed so simple at first is shown to be a more complex, hyperbolic saying that can't possibly be taken literally, in light of related scriptural data.
Jesus couldn’t contradict Himself, nor could His apostles blatantly disregard or be unfamiliar with His teaching (right in inspired Scripture!). Therefore, the objection to calling Catholic priests father must be discarded, as a thoroughly misguided “non-starter.”