This pretty much all boils down to one thing, and one thing only, and that is whether or not society accepts that people are born homosexual (the position that is supported by all recent science). Once this basic principle is established, there is no way, in the long term, that state-sponsored or sanctioned discrimination against homosexual people can stand. The folks who have strong feelings against homosexuality on their own moral grounds are going to fight this to the end, because they know that if this principle is established, they and their institutions are going to be held to the same non-discrimination standards with regard to homosexual persons as they with regard to race, religion, disability, and country of origin.
In short, Catholic hospitals want to be able to discriminate in hiring or firing against someone who is in a same-sex relationship and want to avoid providing same-sex health and retirement benefits on par with married employees. If orientation is elevated to the level of race as a protected class, as I believe it eventually will be as the science continues to prove that homosexuality is an inherent, born trait, they will not be able to discriminate in this way. And that is going to be really, really painful for them.
You can clearly see this fear in the links posted to the sites supporting the amendment. They are clearly identifying this issue as one of the major "harms" that they think will come from allowing same sex marriage. In fact, it is not same sex marriage per se that is going to cause this to come to pass, but rather the societal recognition of what is already pretty established scientific fact - just as society recognized, over the course of the past 150 years, that there is no rational basis for the notion that some "races" of people are inherently inferior to other "races".
Plenty of devout religious people who thought they had strong moral compasses pointed to the Bible 75 years ago to uphold their claims that integration and interracial marriage were evil, and that women were inferior to men and shouldn't be able to vote (St. Paul is very clear on this latter point, I believe in Timothy). Eventually, their "moral code" had to give way to new facts, or rather, facts that had theretofore been unrecognized or denied.