Did he tell you that he has AS, or are you just guessing that?
Speaking from my own experience, he's probably just unsure about how to "approach" (I hate this term for some reason). Just bring up the subject indirectly, like joke about "so why haven't you asked me out yet?" or something. It would work best if you did this right after one of his awkward flirting attempts that you mention, so you're not just bringing it up out of nowhere. He should summon the courage eventually, but since we Aspies tend to overthink everything it's best not to leave any room for doubt he may have (for fear of rejection).
I know I don't speak for every guy, and this doesn't jibe with the so-called "enlightened" perspective of gender that is so common today, but even though I have my own trouble asking girls out I still feel weird about girls asking guys on dates. Feel free to encourage him to death, though. 