No, he isn't.
You do have to read the books to understand that:
a) Ned suffers from PTSD. He's had zero tolerance where the murder of kids is concerned, ever since he saw what Tywin did to the Targaryan kids. This, alongside the guilt of keeping that secret for so long, is what haunts his thoughts. It's not just a moral stance - he's traumatized. This is why he talks to Cersei.
b) Ned isn't interested in playing. You get this vibe from the show somewhat too, but in the show, it's Ned who insists that he must accept Robert's offer, out of a sense of duty. Whereas in the books, it's Cat who pushes him - he didn't want to leave home, and he definitely didn't want to play politics in King's Landing. Nor play at war, for that matter. It wasn't just kids; Ned wasn't keen on bloodshed in general. Honestly though, I don't think this aspect significantly contributes to his downfall - it's almost an non-issue. Tyrion failed because he was too interested in playing the game, too readily making enemies and trusting allies as friends. Tywin failed too. Littlefinger will fail soon. Ned's failure (and his weaknesses) is no bigger than theirs. They're all "naive" in different ways.
Also, I'm not sure he could have gotten "eyes" to rival the spymasters' (or that didn't belong to them outright), in their own territory, even if he wanted to, in the time that he had. He could have had a handful of loyal northerners try their hand at that for him, but I'm not sure how effective/worth doing this would have been.
And to be fair, Littlefinger intended that warning to make him appear more trustworthy. And Ned believing Littlefinger to still be in love with his wife (meaning he would act in her best interests, which were also Ned's) would be a reason to trust him, if anything. The problem with LF was that he wanted revenge for something that happened decades ago, and lusted for power more than his amicable facade might have suggested (this is another difference; book LF hides his malevolence better than in the show).
If anything, I'd say he was one of the wiser characters. And he's definitely not inexperienced as a ruler or soldier. I would call him slightly closed-minded, in that instead of keeping a more open mind, he dismisses Arya's story as a child's ramblings and doesn't consider it beyond that, and basically accepts Littlefinger as an ally because Cat trusts him, without much further consideration.