“Think not that I have come to abolish the law and the prophets; I have come not to abolish them but to fulfil them.” (Matthew 5:17)
We observe indirect indication of Jesus’ acceptance of the traditional Jewish prohibition of homosexual acts, or sodomy, in his approval of the judgment of Sodom (Mt 10:15; 11:23-24; Lk 10:12; 17:29). Obviously, He would agree with that, since, being God the Son, he himself judged Sodom and Gomorrah (Jn 5:22).
Elsewhere in the New Testament it is further explained how the Jews regarded Sodom, and why it was judged:
1 Timothy 1:10 (RSV) immoral persons, sodomites, kidnapers, liars, perjurers, and whatever else is contrary to sound doctrine,
2 Peter 2:6 if by turning the cities of Sodom and Gomor’rah to ashes he condemned them to extinction and made them an example to those who were to be ungodly;
Jude 1:4, 7-8 For admission has been secretly gained by some who long ago were designated for this condemnation, ungodly persons who pervert the grace of our God into licentiousness and deny our only Master and Lord, Jesus Christ. . . . just as Sodom and Gomor’rah and the surrounding cities, which likewise acted immorally and indulged in unnatural lust, serve as an example by undergoing a punishment of eternal fire. Yet in like manner these men in their dreamings defile the flesh, reject authority, and revile the glorious ones.
Genesis 19:4-5 But before they lay down, the men of the city, the men of Sodom, both young and old, all the people to the last man, surrounded the house; and they called to Lot, “Where are the men who came to you tonight? Bring them out to us, that we may know them.”
According to Gesenius’ Hebrew-Chaldee Lexicon to the Old Testament (edition of Baker Book House [Grand Rapids, Michigan], p. 334; Strong’s word #3045: yada):
(3) to know, to become acquainted with any one . . . Often put by a euphemism for sexual intercourse. — (a) of a man; to know a woman, i.e., to lie with her, Gen. 4:17,25; 1 Sa. 1:19, etc.; also as applied to crimes against nature, Gen. 19:5.
The International Standard Bible Encyclopedia (Grand Rapids, Michigan: Eerdmans, 1939), Vol. V, p. 2821 (“Sodomite”), states that “Sodomite” is the English translation of the Hebrew kadhesh (or qadesh): Strong’s word #6945), which referred to male temple prostitutes (Deut 23:17; 1 Kings 14:24; 15:12; 22:46; 2 Kings 23:7).
The King James Version uses “sodomite” for all five of these passages.
RSV: “cult prostitute of the sons of Israel” (Deut 23:17)
“Male cult prostitute” (1 Ki 14:24; 15:12; 22:46; 2 Ki 23:7)
The New Bible Dictionary (Grand Rapids, Michigan: Eerdmans, 1962), notes:
Gn. 19:4-5 concentrates on sexual perversion, particularly homosexuality. (“Plain, Cities of the,” p. 1003)
Another way that can be used to show that Jesus disapproved of homosexual relations is to note what the Mosaic Law stated about it:
Leviticus 18:20-30 And you shall not lie carnally with your neighbor’s wife, and defile yourself with her. You shall not give any of your children to devote them by fire to Molech, and so profane the name of your God: I am the LORD.  You shall not lie with a male as with a woman; it is an abomination.  And you shall not lie with any beast and defile yourself with it, neither shall any woman give herself to a beast to lie with it: it is perversion. “Do not defile yourselves by any of these things, for by all these the nations I am casting out before you defiled themselves; and the land became defiled, so that I punished its iniquity, and the land vomited out its inhabitants. But you shall keep my statutes and my ordinances and do none of these abominations, either the native or the stranger who sojourns among you (for all of these abominations the men of the land did, who were before you, so that the land became defiled); lest the land vomit you out, when you defile it, as it vomited out the nation that was before you. For whoever shall do any of these abominations, the persons that do them shall be cut off from among their people. So keep my charge never to practice any of these abominable customs which were practiced before you, and never to defile yourselves by them: I am the LORD your God.”
Leviticus 20:13 If a man lies with a male as with a woman, both of them have committed an abomination; . . .
And what did Jesus think of this Mosaic Law?:
Matthew 5:17-19 Think not that I have come to abolish the law and the prophets; I have come not to abolish them but to fulfil them. For truly, I say to you, till heaven and earth pass away, not an iota, not a dot, will pass from the law until all is accomplished. Whoever then relaxes one of the least of these commandments and teaches men so, shall be called least in the kingdom of heaven; but he who does them and teaches them shall be called great in the kingdom of heaven. (cf. Mt 7:12; 22:40; Lk 16:17)
Matthew 23:2-3 The scribes and the Pharisees sit on Moses’ seat; so practice and observe whatever they tell you, but not what they do; for they preach, but do not practice. (cf. Mt 23:23)
Jesus observed the Law in its entirety. He worshiped in synagogues and the Temple; he participated in the Jewish feasts (the Last Supper was a Passover dinner: Mt 26:17-19; Mk 14:12-16; Lk 22:7-15). He casually made reference to the authority of Moses over his own disciples (Mt 8:4; Mk 1:44; 7:8-13; Lk 16:31; 24:44; Jn 5:46; 7:19-23).
Apart from the stoning, which was no longer practiced in the new covenant (see his treatment of the woman caught in adultery: Jn 8:3-7), there is no indication that he disputed any of these received laws. Therefore, he accepted as part of the whole, the injunctions against homosexual acts. Case closed.
When the Bible refers to marriage, it’s always between a man and a woman.