The Church has repeatedly supported both Monarchy and the tenets conveniently called Distributism. While this implies that Monarchy is Distributist, how is this specifically the case?
The core concepts of Distributism are
1) Private ownership of property is both good and a right
2) All arrangements, including business and labor arrangements, should be made freely
3) Leadership should be as local and personal as possible; these two thing demand that it also be as small as possible
4) Utopia is impossible.
How does Monarchy meet these criteria while, say, a Republic does not?
Private Property: Private ownership of property is a good because it makes the owner responsible for his property, leading him to learn and grow from its care. An owner's personal interest will lead him to make property more productive, improving society as a whole directly and indirectly. On the other hand, communal ownership of property leads to the Tragedy of the Commons; this is the oft-proven fact that when people own something jointly they eventually deplete it even as they are aware that this depletion hurts everyone, including themselves.
As was pointed out by the Libertarian/Minarchist theorist Hans-Hermann Hoppe in his book 'Democracy: The God that Failed' Democracy is, effectively, the joint ownership of the means of government. He argues, persuasively, that the Tragedy of the Commons is as inherent to Democratic governments as it is to the world's fisheries. Democracy, he argues, is doomed to self-destruction by its very nature.
He goes on to demonstrate that Monarchy is the private ownership of the means of government; the monarch's own self-interest will drive the development of government to be more efficient and more productive overall. While just as a farm may have a bad owner who does not maintain it a kingdom may have a poor king, over time the nature of Monarchies is to improve so long as they do not become democratic and the nature of Democracies is to degrade until they collapse.
Nobles within a Monarchy are the 'local owners'; they are the 'personal owners of the local forms of government' and have the same benefits (vested interests in improving govnerment, etc.) as the monarch.
Free Arrangement: While the most obvious versions of agreements between people and groups are business contracts (private individuals and groups) and treaties (governments) they also include such seemingly-ephemeral ideas as the Social Contract. From Grotius to Rawls theorists have argued that the source of governmental authority has been a social contract existing between the citizen and the leader or government.
When a government is forced to accept a treaty against its will it is typically called a surrender. When a person is forced into labor against their will and are unable to quit, it is typically called slavery. Yet when a person born into a Democratic society is forced to accept an unwritten, implicit social contract with no chance to refuse it this is typically simply not addressed. Proudhon, Hume, Spooner, and other theorists have pointed out that in the absence of actual consent such a 'social contract' is at best a convenient fiction.
Within a Monarchy, however, the contract (or oath of fealty) is explicit and voluntary. This means that the arrangement (or contract, or oath) is made freely and by full consent. Likewise, oaths of fealty with local aristocrats are still explicit and voluntary.
Local and Personal: While Democracy claims to be 'rule by the people' in actuality Democracies fall to Moore's Laws of Bureaucracies, Pournelle's Iron Law of Bureaucracy, and Parkinson's Law. To sum up those theories, over time Democracies become more and more bureaucratic and remote from the people. The reasons for this are simple and inherent to Democracy; politicians are transient but institutions are not, so bureaucracies must exist within a Democracy simply to get things done. As the various 'laws' above point out, over time bureaucracies naturally expand in size, influence, and insulation from outside pressures eventually existing merely to propagate themselves and to increase their own size and power. Eventually the bureaucracies have such influence that the transient politicians are too dependent upon them to reform them; the 'power of the people' is then resident in unelected and unaccountable bureaucrats.
Even before this state of affairs politicians are relatively remote. Their transience means that they are not capable of a permanent attachment to the roles and duties of any office. Indeed, election processes may mean that they are focused on being elected during the entirety of their tenure in office!
Within a Monarchy, however, the costs of a bureaucracy come directly from the wealth of the monarch and nobles, meaning the owners of the means of government have very incentive to avoid and control bureaucracy. Indeed, with a robust aristocracy you can effectively avoid bureaucracy and have direct interactions with the local ruler (petitioning your local baron in person) and effecting change (petitioning the higher lord directly) as opposed to the impersonal interactions of Democracy (interacting with an unelected, unaccountable bureaucracy and only being able to complain via voting in the next election and hoping your vote will be enough to elect a new politician and that the new politicians will be interested in and capable of reforming the bureaucracy).
No Utopia: Monarchies face the normal checks and balances of life; a particularly bad king will be deposed by the nobles, his own family, or the people. Bad nobles face the same as well as removal by the king. But since the wealth and prestige of the monarch and nobles is directly related to how well their subjects are doing self-interest itself ameliorates the likelihood of a poor rulers.
So, as you can see, Monarchy is naturally Distributist.