I cannot see this occurring without the Catholics divesting themselves from these hospitals. If this were attempted it'd surely end up at the Supreme Court. If a public clash between the Obama White House and the Catholic Church came about what would this portend for the 2012 election? Papa Benedict XVI surely would not sit still for this, making his and the Vatican's viewpoint heard loudly.
On another point, can Obama be any more of a blatant hypocrite? He calls for inclusiveness and nondiscrimination yet wishes to dictate to private hospitals what their policies should be.
Remember that these are just words that liberals use as masks. When they say inclusiveness and nondiscrimination it means inclusion in their positions and nondiscrimination against individuals and groups they specify. Everything else is fair game for discrimination. It is not self-consistent and therefore patently illogical and absurd, but then again these are liberals we're talking about.