Ned already made his decision to go to King's Landing to investigate Jon Arryn's death, a man who was like a father to him and King Robert. The letter Catelyn's sister sent also pointed fingers at the Lannisters. Bran's fall only added more fuel to the fire. Ned was going down there regardless. One thing that could have changed if Bran hadn't been pushed is that Catelyn wouldn't have kidnapped Tyrion.
If Catelyn doesn't kidnap Tyrion. Tywin Lannister doesn't antagonize the Riverlands, Jaime doesn't attack Ned, and Robert doesn't die. This also gives Ned more time to seal the deal on Cersei about the kids not being Roberts.
Catelyn meant well but that's a bigger fumble than Bran being pushed out a window.
We aren't arguing which was the bigger fumble though, we're arguing what actually set off the demise of the Starks. Hypotheticals aside, without that (as you even call it) "fuel to the fire" of Bran being pushed, we can't say for certain how things would have played out. All we know is that humpty dumpty made a great fall and from there the dominos fell (
).
but 
meaning the Starks wouldn't know about the incest NOR would they be pushing up on the Lannisters on some 
