May "gay" men be ordained as priests?
Existing priests who may deal with homosexual tendencies yet are committed to a life of chastity and otherwise to the Church’s teachings can certainly continue in their priestly ministry.
Regarding candidates for the priesthood, in December 2016 the Congregation for the Clergy, with Pope Francis’s approval, reaffirmed that men who live a homosexual lifestyle, have “deep-seated homosexual tendencies,” or support the “so-called ‘gay culture’” cannot be admitted to the ministerial priesthood:
In relation to persons with homosexual tendencies who seek admission to seminary, or discover such a situation in the course of formation, consistent with her own Magisterium, “the Church, while profoundly respecting the persons in question, cannot admit to the seminary or to holy orders those who practise homosexuality, present deep-seated homosexual tendencies or support the so-called ‘gay culture.’ Such persons, in fact, find themselves in a situation that gravely hinders them from relating correctly to men and women. One must in no way overlook the negative consequences that can derive from the ordination of persons with deep-seated homosexual tendencies” (The Gift of Priestly Vocation 199, footnotes omitted).
This document reaffirms that, contrary to some reports, the Church doesn't view the term gay favorably, given its association with living and promoting a lifestyle that is irreconcilable with Jesus Christ and his teaching (see John 8:31-32).