For us, however, a key difference is that only conscious reasoning can make use of working memory to hold intermediate conclusions, and accordingly reason in a recursive way (Johnson-Laird, 2006, p. 69): primitive recursion, by definition, calls for a memory of the results of intermediate computations (Hopcroft & Ulmann, 1979). [… example task omitted …] The non-recursive processes of intuition cannot make this inference, but when we deliberate about it consciously, we grasp its validity (Cherubini & Johnson-Laird, 2004). Conscious reasoning therefore has a greater computational power than unconscious reasoning, and so it can on occasion overrule our intuitions.
There’s no evidence that whatever bits of memory intuition uses cannot do recursion. Hunting through semantic memory structures can be viewed as a recursive process and the process is not (at least always) accessible to consciousness. Aside from this, you can impose recursion on just about any process you care to analyse, and you can often remove recursion from a process description depending on what primitives are available. Questioning whether a process “is” or “isn’t” recursive isn’t a healthy activity. Also the jump from “recursive” to “primitive recursive”, as if they were one and the same, is deeply confusing. See the Stanford Encyclopaedia of Philosophy for details of other flavours of recursion.
Bucciarelli, M.; Khemlani, S. & Johnson-Laird, P. N. (2008). The psychology of moral reasoning. Judgment and Decision Making, 3, 121-139