Question: Why doesn't the Orthodox Church recognize the rights of homosexuals to get married?
Answer: Besides the fact that there is absolutely no indication within Scripture or Tradition that marriage is to be anything but between a man and a woman, and that Scripture and Tradition make it absolutely clear that sexuality used in any other way outside the confines of marriage between a man and a woman is worthy of eternal damnation, there is one primary proof text within Scripture that theologically can offer no possibility for there to be marriage outside of a male/female relationship.
Saint Paul explains the "great mystery" of marriage in Ephesians 5:22-33, where he writes:
22 Wives, submit to your own husbands, as to the Lord. 23 For the husband is head of the wife, as also Christ is head of the Church; and He is the Savior of the body. 24 Therefore, just as the church is subject to Christ, so let the wives be to their own husbands in everything.
25 Husbands, love your wives, just as Christ also loved the Church and gave Himself for her, 26 that He might sanctify and cleanse her with the washing of water by the word, 27 that He might present her to Himself a glorious Church, not having spot or wrinkle or any such thing, but that she should be holy and without blemish. 28 So husbands ought to love their own wives as their own bodies; he who loves his wife loves himself.
29 For no one ever hated his own flesh, but nourishes and cherishes it, just as the Lord does the Church. 30 For we are members of His body, of His flesh and of His bones. 31 “For this reason a man shall leave his father and mother and be joined to his wife, and the two shall become one flesh.”
32 This is a great mystery, but I speak concerning Christ and the Church. 33 Nevertheless let each one of you in particular so love his own wife as himself, and let the wife see that she respects her husband.
When we read this passage, there is absolutely no way to interpret it outside of the male/female relationship. The Mystery of Marriage has a symbiotic relationship with the mystery of the marriage between Christ and the Church. The image of Christ is specifically represented as a male in the original Greek (ο Xριστος; masculine noun) and the image of the Church is specifically represented as a female (την Eκκλησιαν; feminine noun). In the holy Mystery of Matrimony, the man is called to reflect the role of Christ in his love for his wife, while the woman is called to reflect the role of the Church in her respect for her husband.
A theological analogy such as this is absolutely impossible outside of the male/female relationship. It is based on this theological analogy that the Orthodox Church calls marriage a Great Mystery and is thus considered one of the great Seven Sacraments (Mysteries) of the Church. By a man and a woman joining themselves in Holy Matrimony through the Church, they are primarily entering a union in which they promise to reflect this theological analogy of the relationship between Christ and the Church, which St. Paul calls "a great mystery" and the Orthodox Church teaches is the ultimate purpose of marriage, even more so than that of bearing children.
Essentially, St. Paul teaches that marriage is undertaken by a man and a woman who are weak natured in order to help each other strive to a yet greater holiness. Though it is better to do this in a celibate state if possible (1 Cor. 7:8-9; Matt. 19:12), the path towards holiness can still be achieved in a marital state by living selflessly towards one another and through living the life of the Church in which they entered this union. For this reason, the spiritual goal of the married and the celibate is the same.
A homosexual marriage between two males or between two females cannot reflect the theological analogy taught by St. Paul, which is the foundation of the Mystery of Marriage in a symbiotic way. The role of the husband and wife completely falls apart, making the attainment of mutual holiness impossible. It also undermines the essential Christian doctrine that there can only be One Christ and One Church, since two husbands analogically reflects two christ's and two wives reflect two churches. With all these essential factors set aside, a homosexual marriage proves to have other motivations that are contrary to ecclesiastical life and its ascetical and mystical theology, since marriage ultimately aims at making our bodies and souls pure vessels of the Holy Spirit.