Littlefinger was one of the primary drivers of the war and the assassination attempt on Bran was a part of it.
Littlefinger had Lysa Arryn kill her husband, Jon Arryn -- Ned Stark's friend, who had been investigating the parentage of the incest-made Lannister children.
Littlefinger implicated Tyrion in the knife carrying, and in the books he did so knowing full well that it was actually Joffrey in order to cause more chaos. As noted above, the person who hired and armed the assassin in the books was Joffrey -- and Littlefinger knew it. In order to impress Robert (who said putting Bran out of his misery would be the right thing to do), Joffrey stole the dagger and hired the assassin.
In the show, however, it is revealed in the Season 7 finale that the knife was Littlefinger's own -- Bran reveals that the knife belongs to him at his trial, which allows Sansa to draw out the implication that the enmity between the Starks and the Lannisters is due to Littlefinger's conniving.
These two threads work to pit the Lannisters and Starks against each other -- allowing Littlefinger to both dispose of one of the two people preventing him from separating Sansa Stark from her family and get himself one step closer to the throne, after betraying Ned and setting the events leading to the war in action.
With this knowledge, Littlefinger's chat with Varus becomes a tacit admission of his hand in starting the war by getting the balls rolling. He admits to wanting the throne and having a hand in Lysa Arryn's widowhood in one response while talking about the throne:
Lord Varys: The Lysa Arryn of chairs. Shame you had to settle for your second choice.
Petyr 'Littlefinger' Baelish: Early days, my friend. It is flattering really, you feeling such dread at the prospect of me getting what I want.
The assassination of Jon Arryn is one of the major reason that Ned Stark went to King's Landing, his suspicions against the Lannisters only further exacerbated by the assassination attempt on Bran.
Littlefinger flat out admits to stopping Sansa's marriage to Loras Tyrell:
Lord Varys: Thwarting you has never been my primary ambition, I promise you. Although, who doesn't like to see their friends fail now and then.
Petyr 'Littlefinger' Baelish: You're so right. For instance, when I thwarted your plan to give Sansa Stark to the Tyrells, if I'm going to be honest, I did feel an unmistakable sense of enjoyment there.
Sansa's being held captive was one of the major points leading to the war, and her marriage to the Tyrells was to be her way out of Lannister captivity.
Given our near-omniscient viewpoint, which Bran clearly has some insight into, we see that, yes, Littlefinger is absolutely at fault in starting the war -- and he did so purposefully, because chaos is a ladder and he intended to climb it.