Not true. The Church teaches that the Mosaic Law was abrogated by Christ's sacrifice. It is in baptism into Him that we participate in the New Covenant and are freed from the twin slaveries of sin and the Old Law.
The freedom obtained in baptism is intended to make room for a new guide or principle. That new guide is Jesus. And nowhere does His testament condemn homosexuality.
Again, not true. There are other New Testament condemnations of homosexual behavior, but St. Paul vigorously condemns homosexual behavior in 1 Corinthians 6:
9 Do you not know that the wicked will not inherit the kingdom of God? Do not be deceived: Neither the sexually immoral nor idolaters nor adulterers nor male prostitutes nor homosexual offenders, 10 thieves, the greedy, drunkards, revilers, robbers—none of these will inherit the kingdom of God. (emphasis added)
(sounds like a Democrat convention, doesn't it?)
There is only one "natural moral law" that stands the test of time and runs as a seamless thread through all cultures.
There is a Natural Law that indeed stands the test of time, but it is curious that this person would cite it. The seamless thread through all cultures does not show any traces of homosexual marriage, for example. It simply does not (and did not) exist as a common or natural practice in any culture. Citing Natural Law does not, therefore, aid the author's case. Why start to do now what has never existed in any other culture, that, according to this person's argument, therefore contradicts natural moral law?
What the author might be straining to do is to support his argument with pagan references. In pagan cultures, particularly in Roman and Greek ancient cultures, homosexuality was not strenuously condemned. I have the impression from reading (Suetonius, for example), that permanent homosexuality was not considered truly normative. It was regarded as an acceptable aberration, as it were.
The norm in ancient times was to have children. When birthrates declined for whatever reasons nations tended to fall (like what is happening in Europe and here).
Homosexuals have committed most of the horrifying mass murders of history. After delving in an activity pleasing to the devil, which is homosexual behavior, they become truly unbalanced without repentence and kill their sexual conquests; in some cases by the scores.