The short answer is no, Genocide of any kind is not only not supported, it is condemned as a mortal sin by the Church, even the Orthodox Church in Ughanda.
It may help to be clear on how Orthodox Christians approach sin. First, because every person inherits some curruption of human nature thanks to the fall of Adam, it is presumed that every person deals with some sort of sin. Thus, Christianity is not interested in the condemnation and destruction of the sinner, no matter what the reason behind it is. Christianity is concerned with the restoration of the sinner to the proper human nature that is God-like and pure and all-loving. That being said, it seems that because of societal trends, certain sins condemned by the Church are being pushed on the world's populus for acceptance, thus Christians get railed for condemning the behavior. The fact remains, though, that the homosexual has one sort of struggle against lust, and the heterosexual has another sort of struggle against lust. Both still have to struggle against lust. Thats why we say, get married once and stay married and do not commit adultery, for that is one sort of chastity, or become a monk and practice that sort of chastity. We do not expect different things from different people, nobody perfectly obeys the Church's teachings, and so everyone has to struggle in cooperation with Christ to find salvation.
Anyone who tells you that he does not need to struggle in the Church has not yet taken ownership of and responsibility for his sins. Anyone who judges someone else because of how much the other sins, or how the other sins, that person judges because he is blind to his own sin and doesn't recognize that his actions affect the whole of society, and murderers, adulterers, thiefs, etc. are sinful in part because of his own negative contributions to society. If our brother sins, it is our sin too, for we are all responsible for each other.
I hope this clears things up a little bit. We do not condemn homosexuals, but we do condemn their sin and compassionately urge them to forgive us our faults and to take responsibility for themselves and begin to repent along with the rest of those in the Church.
