Searle’s claim here seems reminiscent in some ways of Aquinas’s argument for the subsistence of the human soul at Summa Theologiae I.75.2:
Therefore the intellectual principle which we call the mind or the intellect has an operation "per se" apart from the body. Now only that which subsists can have an operation "per se." For nothing can operate but what is actual: for which reason we do not say that heat imparts heat, but that what is hot gives heat. We must conclude, therefore, that the human soul, which is called the intellect or the mind, is something incorporeal and subsistent.
Aquinas has argued earlier for the first claim made in this passage, viz. that intellectual operations do not involve a bodily organ. What he saying here is that this claim entails that that which carries out these operations, the human soul, must “subsist” apart from the body; it isn’t a mere accident or attribute of the body. The reason is the Scholastic principle that agere sequitur esse, activity follows upon being. Heat, as a mere accident or attribute, cannot cause something to be hot; rather it is the substance which has the heat that causes something else to be hot. Similarly, an operation which is not carried out by any bodily organ but which – qua operation rather than substance – cannot exist apart from some substance or other, must inhere in something immaterial.
So, when Searle tells us that immaterial mental properties would have to inhere in something immaterial rather than in a material object like the brain, he is saying something which seems to dovetail with Aquinas’s argument.
Still, things are a bit more complicated than that. For one thing, unlike many contemporary property dualists, Aquinas does not regard consciousness (as contemporary philosophers tend to understand “consciousness”) as immaterial. Rather, it is intellectual activity (grasping abstract concepts, reasoning, etc.) that he takes to be immaterial. Second, Aquinas is not only not a property dualist, he is not (contrary to appearances) a substance dualist either, certainly not in the way that sort of view has been understood since Descartes. Rather, he is a hylemorphic dualist. From a Thomistic point of view, substance dualism, property dualism, materialism, idealism, neutral monism, and all other post-Cartesian theories of the mind presuppose a mistaken and muddleheaded conception of both mind and matter – a conception which (among other things) makes it very difficult for contemporary philosophers even to understand the Thomistic view. As when dealing with Aquinas’s position on other specific philosophical questions, the only way properly to understand what he says about the relationship between mind and body is to situate it within his general metaphysics, which presupposes an understanding of the notions of act and potency, form and matter, substance and accident, essence and existence, analogical predication, etc. I set all this out in Aquinas, with chapter 4 devoted to Aquinas’s psychology and how it differs radically from contemporary substance dualism, property dualism, etc.
Now, some further reading, while you wait for your copy of Aquinas to arrive: First, my essay “Why Searle Is a Property Dualist,” which explains why Searle’s own anti-materialist arguments in philosophy of mind do in fact entail property dualism, despite his attempt to avoid this result. Second, check out David Oderberg’s article “Hylemorphic Dualism” for an overview of the Thomistic position. Third, take a look at Alfred Freddoso’s article “Good News, Your Soul Hasn’t Died Quite Yet” for a discussion of some of the differences between the Thomistic view and the standard modern ones (and why Catholics, especially, should be wary of the latter, including modern versions of dualism).