Though, we should also understand the usefulness of religious hospitals. The staff are paid much lower, many of whom volunteer, and they have provided many indigent families with medical care, they otherwise could not afford before the secular hospital ever existed.
"In 2012, the church operated 12.6% of hospitals in the USA, accounting for 15.6% of all admissions, and around 14.5% of hospital expenses (c. 98.6 billion dollars). Compared to the public system, the church provided greater financial assistance or free care to poor patients, and was a leading provider of various low-profit health services such as breast cancer screenings, nutrition programs, trauma, and care of the elderly."
The Church takes a HUGE financial burden off the US Healthcare System and has been involved in healthcare since the inception of the United States. Further, religious institutions operated most hospitals predating the 19th century. So it's not really an "oxymoron" or a contradiction of duties since history demonstrates otherwise.
And Psycho Yuffe, they do allow the treatment of transgender people, they don't specialize in treatment specifically tailored to transgender people though. If you had a broken arm, they would treat you. Same for other LGBT people. For patients with special LGBT needs, they should seek out help from specific healthcare institutions and doctors that specialize in those needs. These hospitals usual have doctors with a broader fields of medicine. For instance, around 1% or less of the population is transgender and the field of study is fairly new. There are only a handful of specialists that exist for the treatments, and these hospitals can't afford to hire those specialists.
I will say emergency abortion is a topic that should be discussed. Since, the mother's life is at stake, and the option to go to another hospital may not exist. Therefore, these hospitals should have the duty to provide in the best interest of the woman's health. However, if a woman decides to have an elective abortion, then the Church shouldn't have to provide the service.
These healthcare facilities are an integral component of the US healthcare system, they are currently NEEDED given the amount of care they provide with lower wages, saving hundreds of billions of dollars each year. Though, yes, some hospitals don't provide care for HIV medications for instance, ect, they lower the burden of other hospitals in other costs, of which would hike up healthcare costs for LGBT patients at secular hospitals. The existence of these hospitals benefits everyone's healthcare needs.