“Accepted with a joyous heart celibacy radiantly proclaims the Reign of God.” (CCC 1579)
These exchanges are based on actual questions I have received (slightly paraphrased):
1. Everyone is unique. Everyone has unique desires.
Indeed. That's why St. Paul says that men should “lead the life which the Lord has assigned to him, and in which God has called him” (1 Corinthians 7:17, RSV). Latin Rite Catholicism chooses its priests from the group of men who are called by God to celibacy (1 Corinthians 7:32-35). He describes marriage as doing “well” (7:38) and celibacy as “better” (7:38). Latin Rite Catholicism prefers the Pauline “better” state for its priests.
Moreover, the celibacy requirement is applied to only a very tiny number of all humanity (and of all Catholics) who wish to become priests in the Latin Rite: who are already called to celibacy and the priesthood by God.
2. It is said that St. Peter had a wife.
Yes he did. And it's written that St. Paul and Jesus didn't. The disciples had a right to marry (1 Corinthians 9:5), but they renounced that right in order to be Jesus’ disciples. Paul also renounced his right to be remunerated for his missionary work (which he talks about in this same passage). He did so for the sake of the gospel — even though it wasn’t required of him. He went “above and beyond the call of duty.” This is perfectly in accord with the Catholic view. Surely the singleness (or in some cases prolonged separation from family) of Jesus, all the disciples and Paul is not insignificant.
Scripture also teaches that in extraordinary cases, voluntary separation from families and even wives was permissible and praiseworthy:
Luke 18:28-30 (RSV) And Peter said, “Lo, we have left our homes and followed you.”  And he said to them, “Truly, I say to you, there is no man who has left house or wife or brothers or parents or children, for the sake of the kingdom of God,  who will not receive manifold more in this time, and in the age to come eternal life.”
Matthew 19:27 Then Peter said in reply, “Lo, we have left everything and followed you. What then shall we have?” (cf. Mark 10:28)
3. I feel that celibacy is unnatural.
For most of us it is, but not for all men, as Jesus said:
Matthew 19:10-12 The disciples said to him, “If such is the case of a man with his wife, it is not expedient to marry.”  But he said to them, “Not all men can receive this saying, but only those to whom it is given.
 For there are eunuchs who have been so from birth, and there are eunuchs who have been made eunuchs by men, and there are eunuchs who have made themselves eunuchs for the sake of the kingdom of heaven. He who is able to receive this, let him receive it.”
We err if we extrapolate our own opinion and feelings to the entire human race. Not everyone has to (or wants to) be married. Let priests follow their calling from God.
4. A priest can’t be married because he is unable to attend to his flock?
It's not an absolute; simply a matter of practicality and wisdom; no divided interest, as Paul notes (1 Corinthians 7:28, 32-35).
5. How is it explained when the Church sometimes accepts married clergy as converts?
By the saying, “there's always an exception to the rule.”
6. If a young priest meets a beautiful woman at a church function and falls madly in love, how does he proceed?
He gets away from her and prays for strength to resist temptations that might lead to what is contrary to morality and his calling.
7. Psychologically, it may be damaging for a priest to remain celibate.
In extreme cases, priests can be released from their priestly duties and “laicized.” One doesn't construct rules and policies (or decide against them) based on hard cases or extreme cases. We don’t throw the baby out with the bathwater, so to speak.
8. If there are married priests in Eastern Catholicism then the Western assertion that celibacy is better for priests is so much “sound and fury signifying nothing.” The Eastern Church would in that case be a scandal to the rest of the Church, by failing to follow the biblical and Pauline ideal.
This doesn't follow at all. It's a discipline, not a dogma, and can change, and did change in the Latin Rite. It simply didn't change in Eastern Catholicism. Western Catholicism believes, following Paul, that sacrificial renunciation of sex and marriage is a heroic sanctity (in our priests).
Eastern Catholicism has simply chosen not to make that a requirement for her priests (just as the West used to not do so). It believes, also following Paul (1 Timothy 3:2) that priests can be married, like bishops (in Paul's time) could be.
That's not a “scandal” or inconsistency at all. It's simply a disciplinary choice that is different from Latin Catholicism. But even in the East, celibacy of bishops is required, so the same principle as the West is acknowledged, but in a more limited way.
People so often think in rigid either/or categories: “if marriage is good, then celibacy must be bad” or “if celibacy is the ideal, then marriage must be bad, and sex wicked and evil.” None of that is the scriptural / Catholic view.
Again, Paul describes marriage and singleness as “well” and “better”: not “bad” and “good” or “good” and “bad.”
Eastern Catholicism also has aspects of renunciation: just in different ways. For example, it has much more rigid requirements for fasting (which affect many more people).
The Catholic Church is big enough to have disciplinary and liturgical diversity without having to play the child's game of assuming that one way must be superior to the other. East and West prefer different liturgies; likewise, they apply different disciplines regarding priestly requirements. Much ado about nothing…