Ned was willing to plunge the entire realm into war over his decision to put Stannis on the throne - Littlefinger stated it succinctly when Ned asks him to buy out the City Watch and its the reason he later turns against him.

Ned's downfall was that he was short-sighted. That's why he wouldn't take the throne during the rebellion (gotta get my sister!), why he made a terrible Hand (all he did was look for the cause of Arryn's death, not run the realm), and confessed to crimes he didn't commit (anyone could see the Lannisters were going to do whatever they wanted with the Stark girls, no matter what he did). Picking Stannis was a bandaid on a cancer that was spreading across the Realm, but Ned couldn't see it.

In the end, all he wanted to do was go back to Winterfell. Screw the civil unrest, the Lannisters taking over everything and, oh yeah, Others creeping up to the Wall. The dude had no ambition and no ability to make long-term plans. He was a reactionist. A cool one who I hated to see beheaded, but a reactionist none the less. That's why he lost his head.

EDIT: Forgot to mention, refusing to kill Dany was a short-sighted move as well, especially when she was pregnant WITH A MALE CHILD WHO WOULD INHERIT AN ARMY. Any properly ruthless noble would have offed her the moment she married a powerful husband. If she wanted to be a turnip farmer somewhere, leave her be. She was clearly being poised for a takeover, but again, Ned couldn't see it.