Whatever one might want to read into Genesis 2:24, God made his intent clear when He codified it in the Law given to Moses:
22Thou shalt not lie with mankind, as with womankind: it is abomination.
Leviticus 18:22 (KJV)
Such an act was so grievous an attack on the general health and welfare of the community that it attracted the highest possible penalty:
13If a man also lie with mankind, as he lieth with a woman, both of them have committed an abomination: they shall surely be put to death; their blood shall be upon them.
Leviticus 20:13 (KJV)
The LORD didn't create automatons, but beings with the same capacity He, Himself, has: to pursue the desires of the heart. However, He would have been an irresponsible father had He not made it abundantly clear how He would have His children behave. His intent regarding His Law is unmistakable:
19I call heaven and earth to record this day against you, that I have set before you life and death, blessing and cursing: therefore choose life, that both thou and thy seed may live:
Deuteronomy 30:19 (KJV)
Anything prohibited by the Law of the LORD will be seen, by those who have regard for the Law, as something that will impact their life in the land in a negative way. Same-sex coitus is clearly prohibited by the Law of the LORD, and if one's desire is to please Him in preference to oneself, then how one ought to behave in regard to it is a no brainer.
So, yes, as far as the LORD is concerned, and anyone else of a like heart (whose desire is to live and prosper in the land), Genesis 2:24 speaks exclusively to male-female copulation.
It has been suggested in a comment by the OP that I have missed the point of the question.
However, nowhere else in Scripture is the expression "one flesh" used in regard to any other union than that of "a man and a woman". The expression is exclusive.
Jesus, prefacing his answer to the Pharisees' question concerning divorce, said:
4... Have ye not read, that he which made them at the beginning made them male and female, 5And said, For this cause shall a man leave father and mother, and shall cleave to his wife: and they twain shall be one flesh
6Wherefore they are no more twain, but one flesh. What therefore God hath joined together, let not man put asunder.
Matthew 19:4-6 (KJV)
So, when a man decides to leave his father and mother and "cleave to his wife", then they are no longer two, but one.
Jesus goes on to say, "What therefore God hath joined together, let not man put asunder. So the "one flesh" union, i.e. the joining of a man (male) and his wife (female), is an act of God.
So, what constitutes "the union"? The article on Marriage in Judaism at the Jewish Virtual Library has this to say,
Mishnah Kiddushin 1:1 specifies that a woman is acquired (i.e., to be a wife) in three ways: through money, a contract, and sexual intercourse. Ordinarily, all three of these conditions are satisfied, although only one is necessary to effect a binding marriage.
For those who might be distracted by the words "through money", I would suggest following the link and reading the whole of the article.
So, in the opinion of the Rabbis, coitus between a man (male) and a woman (female) is only one means by which a binding marriage might be effected. Nevertheless, in the eyes of the LORD, as expressed by Jesus, once a man takes a woman as his wife, "What therefore God hath joined together, let not man put asunder."
So, reiterating my previous conclusion: as far as the LORD is concerned, and anyone else of a like heart (whose desire is to live and prosper in the land), Genesis 2:24 speaks exclusively to male-female copulation.
Since man in not an automaton and has a God-given capacity to pursue his own desires, he is free to engage in whatever activities to which he might be inclined. However, to pursue desires that are contrary to what the LORD has approved, and teach others to do so, will have very predictable consequences:
26Ye shall therefore keep my statutes and my judgments, and shall not commit any of these abominations; neither any of your own nation, nor any stranger that sojourneth among you: 27( For all these abominations have the men of the land done, which were before you, and the land is defiled;) 28That the land spew not you out also, when ye defile it, as it spewed out the nations that were before you.
Leviticus 18:26-28 (KJV)