Our internet friend Dr. X seems to raise two questions about religious charitable organizations to which I will try to respond as an "amateur bloviator."
His first is to ask, what about, say, Muslim patients who refuse to see other-sex doctors (and but hey - what about the nurses and the gay docs?)
Second, he considers whether it is up to the government, via the Big Government payors Medicaid and Medicare or (slightly indirectly) Obamacare, to decide what they want to pay for.
As for the first question, no problem. They should get their care from whomever they choose in whatever form they choose (unless trying to die in the ER). It's called freedom. (Last I read, however, Muslims may be waivered from ObamaCare anyway because it's not Sharia or whatever reason.)
As for the second question, of course government payors can decide what to be willing pay for, given whatever Congress, the bureaucracy, and etc decides. However, that does not, or should not, require that it be provided. The requirement is the rub, and it constitutes a federal takeover, an overreach, an intrusion into choice.
Of course, the larger issue is the politicization and governmentalization of medical care, which promises to create endless explosions if ObamaCare proceeds. Dr. X is quite right that we are headed towards turning the corner where medical coverage is no longer insurance, but plain payment with the feds as the Grand Medical Commission in DC.
Rather than ObamaCare, I would have liked to have seen a wide-open, nation-wide market for private (including charitable) medical coverage of every size, shape, and sort with no federal involvement, and subsidies for the poor of any age - old or young. (Medicare was a giant error, since the elderly, statistically, are wealthier than the young who pay those bills with their taxes.) Muslim policies for Muslims if they want them, Muslim clinics and hospitals if they want them, Catholic policies if Catholics want to buy them, etc. etc. If unrestrained by government, the market would be providing for every individual or family want. Unfortunately, we never had a wide-open market for medical insurance due to the state insurance commissions' protectionism, or whatever it has been.