No, government occurs when people whose will is to be enforced by men with guns decide what shall be done with your wealth, your energies, and ultimately your thoughts.
The entire article is built upon the false equation of government with society (the latter, not the former, is comprised of institutions such as marriage and family), and attempts to attach the good name of the latter to the former. The good deacon disguises this trick better than most, but it's been used so often that perceptive people are on the watch for it.
Once those principles are conceded and firmly established, you might be surprised at how much many (most?) very religious people will be happy to live in a quite libertarian society. As a libertarian might wish to devote his charitable contributions to freedom libraries so a born-again may wish to tithe to the religious congregation to which he/she belongs. Neither should have objection to what the other wishes to subsidize in such a way. Neither has an interest in supporting government taxing anyone to the point of being unable to contribute as he/she sees fit.
If I am not mistaken, Deacon Keith Fournier (few Catholic deacons run around seeking attention by calling themselves Deacon as casually as physicians call themselves Doctor so be on an agenda watch for Fournier) was originally of the infamous Archdiocese of Boston and was then employed by Rev. Mr. Pat Robertson to run a Catholic affiliate of Christian Coalition. He also had been involved as a functionary in the Massachusetts Demonratic Party but was understandably fed up with abortion. He is a voice without much of a constituency in that he is a raving economic and governmental economic liberal while being socially conservative. His Democrat Party died in about 1960 but he was not old enough to notice.
Traditionally, Americans recognized, unlike Fournier, the wisdom of Jefferson in his estimation that, while government may be necessary, that government governs best which governs least. We are sometimes tempted to the obvious conclusion: that the government which governs not at all governs best of all. Government (with guns) has obligations to protect us from enemies foreign and domestic: enemy powers bent on our destruction and criminals (both of which are initiators of force in libertarian speak). As a Catholic and as a former state Libertarian Party Secretary who turned to the GOP, I would say that it is wise not to take the principle too far.
It may well be that Pope Leo XIII's brilliant encyclical Rerum Novarum (variously translated as the literal Of New Things or the figurative On the Condition of the Working Class) is the model that Fournier ought be looking for and that all of us, Catholic or not, believers or not, might benefit from, speaking in libertarian as well as Catholic terms. That Fournier does not understand the connection between the knowledge of Leo XIII and that of von Mises and von Hayek does not excuse the rest of us.