There's some truth to what he said, but only some.

Let's turn back the clock to the 19th century and the industrial revolution. It really wasn't pretty. I'm sure you know all of the basics here with the horrible pollution and working conditions, child labor, black lung and mangled arms and all the rest.

This of course inspired widespread horror for obvious reasons and not just on the left. A lot of people on the right didn't like early capitalism much either because it disrupted old traditions, turned every relationship into something based on money, made a lot of the old landed gentry irrelevant, etc. These days you have a lot of people on the right who will proudly call themselves capitalists, which really wasn't the case back then.

At the same time a lot of people hated socialism and hated it a lot. It didn't help that many strains of socialism (including the increasingly dominant Marxism) were strongly anti-religious.

So if you're horrified by capitalism and terrified of socialism what do you do? Well you look for a Third Way. There's been a long history of people calling for a "third way" (using that exact term but meaning different things by it) that marked a middle path between socialism and capitalism from fascists to moderate Tories to reformist Eastern Bloc communists to Bill Clinton to the Pope.

Specifically Pope Leo XIII in the Rerum Novarum encyclical of 1891. In Catholic social thought Rerum Novarum is huge and it's certainly what Douthat is referring to. Here's the complete text of it translated into English, it's not too long: https://www.cctwincities.org/wp-content/uploads/2015/11/Rerum-Novarum.pdf

Here's an important bit:

"19. The great mistake made in regard to the matter now under consideration is to take up with the notion that class is naturally hostile to class, and that the wealthy and the working men are intended by nature to live in mutual conflict. So irrational and so false is this view that the direct contrary is the truth. Just as the symmetry of the human frame is the result of the suitable arrangement of the different parts of the body, so in a State is it ordained by nature that these two classes should dwell in harmony and agreement, so as to maintain the balance of the body politic. Each needs the other: capital cannot do without labor, nor labor without capital. Mutual agreement results in the beauty of good order, while perpetual conflict necessarily produces confusion and savage barbarity. Now, in preventing such strife as this, and in uprooting it, the efficacy of Christian institutions is marvellous and manifold. First of all, there is no intermediary more powerful than religion (whereof the Church is the interpreter and guardian) in drawing the rich and the working class together, by reminding each of its duties to the other, and especially of the obligations of justice."

The basic idea is that instead of having capitalists do whatever they want or workers revolt, both sides should cooperate for the betterment of both. This means that both sides have rights and duties. Workers have to work faithfully but employers have to let the workers have time off to go to church and not subject them to degrading poverty and working conditions. Importantly it says that it is unjust for the employer to just give the lowest wage he can get away with because people are desperate and to treat workers with dignity.

And then the Pope goes farther, he specifically says that the state should get involved to make that happen:

"36. Whenever the general interest or any particular class suffers, or is threatened with harm, which can in no other way be met or prevented, the public authority must step in to deal with it. Now, it is to the interest of the community, as well as of the individual, that peace and good order should be maintained; that all things should be carried on in accordance with God's laws and those of nature; that the discipline of family life should be observed and that religion should be obeyed; that a high standard of morality should prevail, both in public and private life; that justice should be held sacred and that no one should injure another with impunity; that the members of the commonwealth should grow up to man's estate strong and robust, and capable, if need be, of guarding and defending their country. If by a strike of workers or concerted interruption of work there should be imminent danger of disturbance to the public peace; or if circumstances were such as that among the working class the ties of family life were relaxed; if religion were found to suffer through the workers not having time and opportunity afforded them to practice its duties; if in workshops and factories there were danger to morals through the mixing of the sexes or from other harmful occasions of evil; or if employers laid burdens upon their workmen which were unjust, or degraded them with conditions repugnant to their dignity as human beings; finally, if health were endangered by excessive labor, or by work unsuited to sex or age - in such cases, there can be no question but that, within certain limits, it would be right to invoke the aid and authority of the law. The limits must be determined by the nature of the occasion which calls for the law's interference - the principle being that the law must not undertake more, nor proceed further, than is required for the remedy of the evil or the removal of the mischief."

To summarize all that, the Pope says it's the job of the government do put down labor unrest but to also stop capitalists from treating people terribly. Of course this isn't just the Pope on a mountaintop shouting out his own ideas, these sort of ideas were widespread beforehand which is why Leo XIII was able to get away with saying all of this and why many people were willing to listen.

So this right here is a blueprint for all kinds of Catholic-oriented political movements and Rerum Novarum had widespread and persistent influence. However if you read the actual text it's not talking about "the state should give people benefits" but rather "the state should force labor and capital to play together nicely." But Rerum Novarum did lead to the kind of reformist Catholic-oriented political movements that contributed a lot to European welfare states.