I agree that the Supreme Court is likely to rule that marrying someone of the same gender is a right guaranteed by the US constitution when the issue goes before the bench next term. The breakdown will likely look like this: Justices Kagen, Sotomayor, Ginsburg, Breyer, and Kennedy in favor of the ruling, with Justices Alito, Thomas, Scalia, and Chief Justice Roberts against. The ruling will probably be very narrow. Justice Kennedy will likely author the majority opinion, and Scalia will write a scathing dissent, even if the other justices do as well.
Of course, if there's any doubt in how this case will go, it's in how Justice Kennedy will rule. He's notoriously quiet during oral arguments, which makes it hard to tell what he's thinking. He's a centrist on nearly all issues, with the notable exception of campaign finance.
As for your question, it really all depends on how "broadly" or "narrowly" the court rules when it makes its decision. When the Supreme Court makes a decision it doesn't just pick a side, it decides how much and to what extent its ruling applies. For example, if the Supreme Court ruled against some kind of campaign finance law in a constitutionality case, they would have to decide how broadly to rule. Does the ruling apply only to federal officers or all elected officials? That's a very simple example (they would almost certainly apply that decision to all elected officials), but you get the idea. When this issue comes up, if the Supreme Court rules using language that refers to gay marriage as a "civil right," or says that gay marriage must be treated equally by all institutions - not just the government - it is likely the Church would need to significantly restructure its operations in the United States.
That is highly unlikely to happen in this case. This Supreme Court has been very cautious and typically rules fairly narrowly. They've also recently shown a special concern over religious issues, as evidenced by their ruling in Hobby Lobby v. HHS. They ruled that RIFRA - the religious freedom restoration act - protected the rights of the CEO of Hobby Lobby not to pay for contraceptive coverage he had a moral objection to. RIFRA prevents the government from passing laws that require someone (such as a preacher preforming marriages) to violate their religious beliefs unless the government can prove that it has a "compelling interest" in passing the law, and that they've chosen the "least restrictive means" of doing so. If Congress tried to force preachers to preform the wedding ceremonies for same-sex marriages, I think they would have an extremely difficult - if not impossible - task in proving it had chosen the "least restrictive means" of allowing same sex couples to marry. After all, what's stopping gay couples from marrying at the local courthouse? They would reap all the same benefits - at least from a legal standpoint - that a traditional couple would.
That having been said, I don't think your hypothetical scenario is too far down the road. There has been a significant degradation of religious rights in favor of perceived "civil rights" over the past 5 years. In fact, Senate Democrats were so angry over the Supreme Court's decision to protect the religious rights of the CEO of Hobby Lobby instead of women's rights to birth control, they attempted to weaken RIFRA to allow the contraceptive mandate to continue. It was mostly a political gesture (they knew the Republican-controlled House would never pass the bill), but the sentiment was real.
Although citizens who are affiliated with religious institutions haven't really been targeted yet, many businesses supplying wedding services have been. For example, a wedding photographer in New Mexico was sued by a gay couple after she refused to photograph the couple's commitment ceremony. The gay couple ultimately won. The Washington State Attorney General sued a florist for civil rights violations after she refused to provide flowers for a wedding. In a more recent case, a gay teacher at a private Catholic school was fired after announcing plans to marry his partner. The teacher is now suing the school for wrongful termination.
In all of these cases, the person or institution involved said their actions were based on religious belief. These are just a handful of many such cases playing out across the country.
So I don't think your hypothetical is out of the question. I think it's only a matter of time before it becomes more common for gay couples to sue religious institutions directly. Chances are, they would win. However, that's more likely to come a few years down the road. It's very, very unlikely that it would come as a result of the upcoming Supreme Court case.
Side note: For readers who are concerned (as I am) about this degradation of religious freedom, you may want to look at the Church's recent public campaign effort to support religious liberty. They've created a Facebook page, and published a series of articles on their newsroom website. You might also want to check out the Becket Fund for Religious Liberty, one of my favorite legal organizations. They take no position on things like gay marriage and abortion, but support the religious liberty of those who oppose them. They do tons of litigation work, much of which is for religions that aren't Christian. Elder Oaks accepted an award from the organization last year, and the Church has sent apostles to give invocations/benedictions at Becket Fund events. Becket Fund lawyers have also participated in Church presentations on religious liberty.