The short answer, I think, is 'no'. This argument for empiricism mistakenly conflates (i) the justificatory basis for a belief, with (ii) the presuppositions or prerequisites that must be met in order for one's justification to not be defeated.
(Another common version of this mistake is to claim that a priori justification is impossible because experience is required to acquire the concepts with which we think in the first place. It may well be true that experience is a prerequisite for concept-possession and hence thought. But that does nothing to undermine the possibility of a priori justification, i.e. the claim that the basis for believing P needn't include any reference to experience. Experience may play an essential role in belief-formation, without it thereby playing an essential justificatory role. But I'll return to the more subtle mistake, since that is more interesting.)
I discussed this issue further in the last section of my post 'Arguing with Eliezer'. But I'll reproduce it here, since the comments to that post focused on other issues, and I'd be interested to hear what others think of this one...
When "pure thought" tells you that 1 + 1 = 2, "independently of any experience or observation", you are, in effect, observing your own brain as evidence.
It's just fundamentally mistaken to conflate reasoning with "observing your own brain as evidence". For one thing, no amount of mere observation will suffice to bring us to a conclusion, as Lewis Carroll's tortoise taught us. Further, it mistakes content and vehicle. When I judge that p, and subsequently infer q, the basis for my inference is simply p - the proposition itself - and not the psychological fact that I judge that p. I could infer some things from the latter fact too, of course, but that's a very different matter.
In discussion, Eliezer emphasized the demands of (what I call) 'meta-coherence' between our first-order and higher-order beliefs. If you reason from p to q, but further believe that your reasoning in this instance was faulty or unreliable, then this should undermine your belief in q. I agree that reasoning presupposes that one's thought processes are reliable, and a subjectively convincing line of thought may be undermined by showing that the thinker was rationally incapacitated at the time (due to a deceptive drug, say). But presuppositions are not premises. So it simply doesn't follow that the following are equally good arguments:
(1) P, therefore Q
(2) If I were to think about it, I would conclude that Q. Therefore Q.
(Related issues are raised in my post on 'Meta-Evidence' [update]. See also my argument for the inescapability of a priori justification.)