Even if we ignore everything unique about Christian marriage, its status as sacrament is a prima facie objection to the possibility of Christian divorce. Christian churches, as far as I can tell, have never raised the possibility of a formal process that could undo a sacrament – there is not even a name for what, say, “unbaptizing” would be. Among Catholics and Orthodox this problem is even more acute by being repeated in several other sacraments. What would “unconfession” be? Or unconfection of the Eucharistic species? True, there is some sort of dispensation from Holy Orders, but the priesthood as sacrament is seen as indelible.
Any notion of Christian divorce either has to see it as non-sacramental or has to give some account of unsacramenting. Option 2 is as unknown both to history as to spellcheck, and so the possibility of Christian divorce rests on denying it’s a sacrament. Some Reformation churches did this long ago, but it is not a viable option for Catholics.
Catholicism seems to be in the process of looking for a tertium quid between marriage and divorce: a sort of yet-to-be-defined pastoral status that nullifies the binding character of the vow while still not recognizing any formal process like divorce. This new status, as far as I can tell, would not be the result of any one-size-fits-all process but of personal consultation and spiritual guidance but would still have to somehow unsacrament a previous marriage. While I get the reasons for wanting this to be so, the cost to our understanding of the sacraments is usually not factored into the discussion. The cost in question is pretty clear: even if all of sacraments depend on ministers to be accomplished, they are chiefly the exercise of God’s fidelity to the Church, which leaves us with no account of unsacramenting that avoids imputing infidelity to God. This is one of the main reasons why marriage is the paradigm for understanding all of God’s relations to his Church, and why the tradition of the Church saw marriage as indissoluble and as sacramental in a mutually-implicating way.