There are, in general, two ways to to respond to this kind of competition: (i) by offering customers more of what they want or (ii) by harming competitors. The former is good for consumers while the latter is not, and the antitrust laws try to distinguish between these cases with what is called the "no business sense" test. If an action by a firm with market power harms competitors in a way that doesn't make any business sense but for its anticompetitive effect, then it may face prosecution.
I was reminded of this principle when I saw (WSJ link) the leader of the Episcopal church respond to congregations that want to affiliate with a rival Anglican sect, by saying that she'd rather have these properties become Baptist churches or even saloons than continue as sanctuaries for fellow Anglicans.
When the Church of the Good Shepherd in Binghamton, N.Y., left the Episcopal Church ..., the congregation offered to pay for the building in which it worshiped. In return the Episcopal Church sued to seize the building, then sold it for a fraction of the price to someone who turned it into a mosque.
I include this under the heading of "Managing vertical relationships" because the buildings are an "upstream" input into the production of Church services. The Episcopal Church is trying to raise the cost of rivals for accessing this input.
The success of the strategy depends on how much rival costs go up without access to these particular land parcels. Using this criterion, the chances of success seem small.
UPDATE: In response to the very good comments to the post, here is a brief commentary on the "no economic sense" test taken from the short-lived Section 2 Report (since renounced) that described how the Department of Justice would enforce this part of the antitrust laws.
The Department finds the no-economic-sense test useful, among other things, as a counseling device to focus businesspeople on the reasons for undertaking potentially exclusionary conduct. At the same time, the Department does not believe that a trivial benefit should protect conduct that is significantly harmful to consumers and the competitive process. Therefore, the Department does not believe that this test should serve as the general standard under section 2.
So even though it may have been the intent of the Episcopal church to monopolize, its ability to do so is unlikely.