What is the Church's position on priests who publicly identify as gay and are living a celibate life?
First, I think you mean “chaste” rather than “celibate.” Celibacy denotes refraining from getting married, whereas chastity in the case of someone who is single, such as a Catholic priest, refers to total continence, i.e., abstaining from any sexual activity. In the case of a Catholic who is celibate, the two terms should go together, but that isn’t necessarily the case in practice.
While the Church has no formal policy on this matter, a priest who publicly identifies as “gay” yet pledges to lives chastely and also support the Church’s teaching otherwise will likely continue in his priestly ministry.
Such a case occurred with Fr. Gregory Greiten of the Archdiocese of Milwaukee, who in December 2017 announced to his parish congregation, “I am Greg. I am a Roman Catholic priest. And, yes, I am gay!”
In an official statement responding to Fr. Greiten’s announcement, Archbishop Jerome Listecki of Milwaukee said, “We support Fr. Greiten in his own, personal journey and telling his story of coming to understand and live with his sexual orientation. As the Church teaches, those with same-sex attraction must be treated with understanding and compassion. As priests who have made a promise to celibacy, we know that every week there are people in our pews who struggle with the question of homosexuality."
Fr. “Greg’s own story reminds each of us of God’s call to continue to grow in understanding and to live holy, chaste lives,” Listecki added.
The Church has never endorsed use of the word gay in any of its official documents, because it doesn’t believe a person’s identity should be reduced to their sexual inclination, especially in the case of those dealing with same-sex attraction (SSA) and who thus, in contrast to men and women who can truly marry, have no moral recourse to express themselves sexually.
For anyone dealing with SSA and desiring to live a joyfully chaste life, we heartily recommend the work of the Catholic apostolate Courage.