Can a Catholic hold the position that some homosexual priests should be punished by the death penalty? Can Catholics hold the position that some homosexual (=any sexual relation to the other man) priests should be punished by the death penalty? (What about homosexuals in general?)
In his Book of Gomorrah, the Doctor of the Church St. Peter Damian

notes the severe penalties historically attached to such offenses in the Scriptures as well as the canons of Church councils, and argues that such penalties should be even stronger for members of the clergy, who are to be held to a higher standard than the laity. The strongest punishments in the Church’s historic legislation are reserved for those who abuse children and adolescents. They are to be “publicly beaten” and humiliated, “bound in iron chains, worn down with six months of imprisonment, and three days every week to fast on barley bread at sundown,” and confined to a monastery in perpetuity under constant guard.

The Book of Gomorrah: St. Peter Damian's Struggle Against Ecclesiastical Corruption, introduction by translator Hoffman
St. Peter Damian writes in ch. 6 of his Book of Gomorrah that such priests are worthy of death (cf. Rom. 1:32):

May idle prelates of clerics and priests hear! May they hear, and although they might be secure from personal guilt, may they fear themselves to be participants in the guilt of others! Undoubtedly, those who turn a blind eye to the sins of their subjects that they are obligated to correct, also grant to their subjects a license to sin through their ill-considered silence. May they hear, I say, and wisely understand, that all are uniformly worthy of death, indeed, “not only they that do them, but they also who consent to them that do them.”

But, as Hoffman notes (fn. 179),

Damian is not, as some hostile commentators have claimed, recommending the death penalty for sodomy. He is quoting part of Romans 1:29–32, in which the St. Paul lists a large number of sins, including pride, disobedience to parents, dissoluteness, contumely, avarice, sodomy, and others, and concludes that “they who do such things, are worthy of death: and not only they that do them, but they also that consent to them that do them.” The reference is to the gravity of the sin, not a recommendation for capital punishment by the state.

St. Thomas, commentating on Rom. 1:32, seems to agree it is a spiritual (not corporal) death, because he cites Rom. 6:23: &quot;The wages of sin is death.&quot;
However, Pope St. Pius V's 1 April 1566 constitution Cum primum treats clerical sodomites similarly to incorrigible heretics:

§. 11. Si quis crimen nefandum contra naturam, propter quod ira Dei venit in filios diffidentiæ, perpetraverit, Curiæ sæculari puniendus tradatur, &amp; si clericus fuerit, omnibus ordinibus degradatus simili pœnæ subiciatur.
[Engl. transl.:] §. 11. If someone commits that nefarious crime against nature that caused divine wrath to be unleashed against the children of iniquity, he will be given over to the secular arm for punishment [of death?*]; and if he is a cleric, he will be subject to the same punishment after having been stripped of all his degrees [of ecclesiastical dignity].

*I don't know what the secular punishment for sodomy was at the time, but, like with heretics, it doesn't seem the Church is opposed to capital punishment of clerical sodomites.
What he says is reminiscent of St. Thomas saying that excommunicated, obstinate, incorrigible heretics should be delivered &quot;to the secular tribunal to be exterminated thereby from the world by death&quot;.