NEW YORK — Allegations that disgraced ex-Cardinal Theodore McCarrick engaged in sex with adult seminarians have inflamed a long-running debate about the presence of gay men in the Roman Catholic priesthood.

Some conservatives are calling for a purge of all gay priests, a challenging task given that they are believed to be numerous and few are open about their sexual orientation. Moderates want the Church to eliminate the need for secrecy by proclaiming that gay men are welcome if they can be effective priests who commit to celibacy.

Among the most outspoken moderates is Father James Martin, a Jesuit priest and writer whose book, Building a Bridge, envisions a path toward warmer relations between the Catholic Church and the LGBT community.

“The idea of a purge of gay priests is both ridiculous and dangerous,” Martin said in an email. “Any purge would empty parishes and religious orders of the thousands of priests (and bishops) who lead healthy lives of service and faithful lives of celibacy.”

That outlook infuriates some conservative Catholics.

Citing McCarrick’s case, Michael Hichborn, president of an institute that promotes traditional Catholic teaching, says there must be a “complete and thorough removal of all homosexual clergymen from the Church.”

“It is going to be difficult and will likely result in a very serious priest shortage,” Hichborn said. “But it’s definitely worth the effort.”

While the McCarrick scandal has intensified debate in the U.S. about gays in the priesthood, it’s a global issue. Recent gay priest sex scandals have surfaced in Chile, Honduras, France and Italy.

In the U.S., where investigations may determine if Church leaders turned a blind eye to McCarrick’s penchant for young seminarians, there have been follow-up allegations of sexual misconduct in seminaries. Cardinal Sean O’Malley of Boston recently announced an investigation into his diocesan seminary.

Catholic teaching, when it comes to homosexuality, is nuanced. The Church says gays should be treated with dignity and respect, yet it has long taught that homosexual acts are “intrinsically disordered.”

In 2005, the Vatican stated that even celibate gays should not be priests, saying Church leaders cannot accept seminary applicants who “practice homosexuality, present deep-seated homosexual tendencies or support the so-called ‘gay culture.'”

Pope Francis has reaffirmed this policy, despite his famous “Who am I to judge?” comment in 2013 when asked about a purportedly gay priest.

In a May meeting with Italian bishops, Francis said, “If there’s even the slightest doubt, better not to accept them” into seminary, according to participants at the closed-door session.

On the front lines in implementing that policy are priests like Father Thomas Berg, admissions director at St. Joseph’s Seminary in Yonkers, New York.

Berg said he and his colleagues strive to rigorously screen the young men applying for admission, assessing their psychosexual development and emotional maturity. Applicants are asked about their dating history and their level of attraction to other males; Berg believes the process has reduced the number of seminarians with same-sex attraction.

As for gays already serving as priests, Berg says he doesn’t advocate a “witch hunt” to root them out. But he says the Church needs to identify sexually active priests, challenge them to repent, and consider their removal from the priesthood.

Berg proposes that dioceses appoint independent watchdogs — ideally people with law enforcement background — to receive and assess anonymous allegations of clergy sexual misconduct.

“Our problem is sexually active priests who are breaking their commitment to celibacy,” Berg said. “That wreaks havoc.”

Francis DeBernardo of New Ways Ministry, which advocates equality for LGBT Catholics, questioned the effectiveness of the seminary screening process.

“Anecdotally, what we’re finding is that the policy encourages people to lie,” DeBernardo said. “If a man feels called to the priesthood, he’ll rationalize that he should not admit his sexuality.”

The effort to exclude gays complicates things for those who do become priests, DeBernardo added.

“The institutional leaders want to promote a message that gay men should not exist in the priesthood,” he said. “So they don’t offer healthy, holy examples of gay priests who are living their celibacy in effective ways.”

Rome-based journalist Robert Mickens, a veteran of Vatican coverage, argued in a recent essay that the Church should be more forthright in acknowledging the substantial presence of gay priests.

“Rather than encourage a healthy discussion about how gays can commit themselves to celibate chastity in a wholesome way, the Church’s official policies and teachings drive such men even deeper into the closet,” Mickens wrote.

Some conservative Catholics blame the climate of secrecy directly on gay clergy, contending there is a “homosexual subculture” in many dioceses and seminaries.

“Numerous reports from clergy and seminarians are coming out worldwide which confirm the existence of networks of homosexually active men who cover for each other,” said Father Paul Sullins, who has taught sociology at Catholic University in Washington.

The current debate over gay priests is framed by the allegations against McCarrick — that he allegedly had sex with adult seminarians as well as abusing minors. Francis ordered him removed from public ministry in June.

In past years, the debate has often focused on the problem of child sex abuse by priests — and the extent to which homosexuality played a role. Those questions are being revisited following the recent release of a grand jury report in Pennsylvania detailing alleged sexual abuse of more than 1,000 children by about 300 priests in six dioceses over a 70-year period.

A study by the John Jay College of Criminal Justice in New York, released in 2011, concluded that gay priests were no more likely than straight priests to abuse minors. Some conservatives, noting that about 80 percent of the abuse victims were male, nonetheless cite the findings to advocate for a purge of gay priests.

One of the top conservatives in the U.S. Catholic leadership, Cardinal Raymond Burke, indicated this month that he favors at least a partial purge.

“Now it seems clear in light of these recent terrible scandals that indeed there is a homosexual culture, not only among the clergy but even within the hierarchy, which needs to be purified at the root,” he said in an interview with Catholic Action for Faith and Family, a conservative advocacy group.

“What is needed is an honest investigation into the alleged situations of grave immorality followed by effective action to sanction those responsible,” Burke said. “Shepherds can go astray … and then must be appropriately disciplined and even dismissed from the clerical state.”

One of Burke’s moderate colleagues, Cardinal Blase Cupich of Chicago, sounded a different tone in an interview with the Jesuit magazine America.

“I really believe that the issue here is more about a culture of clericalism in which some who are ordained feel they are privileged and therefore protected so that they can do what they want,” Cupich said. “People, whether heterosexual or homosexual, need to live by the Gospel.”

Associated Press writer Nicole Winfield, who covers the Vatican for the AP, contributed to this report.