Toy Story? The Matrix? Jurassic Park? Frozen? I think they all qualify, especially the first three.
Just to be clear, I agree with you that the first three qualify as films having a big cultural impact on their own. But still, there are a few things responsible for their bigger cultural impact that have less to do with the film itself, I think.
First,
all of them are now at least trilogies. Yes, I would agree with the stance that every one of them made a lot of impact on their own (and maybe, for Matrix and Jurassic Park, the sequels didn't any good), but still, if you have three films, it's definitely easier to stay in the mind of the public, especially if there's a bit of time between the original and the sequels.
Then there's the fact that Toy Story is Disney. If there's any company that knows how to produce cultural impact, it's them. Just ask people how many of the old stories they adapted are thought to be original stories by them (yes, that's mainly older films, but I'm just trying to emphasise how good Disney is at this). As to merchandise, I mean, it's a film about
living toys. Merchandise practically produces itself.
And Jurassic Park? Come on. It has
dinosaurs in it. The one thing people know about Earth's prehistory. JP certainly helped the dinosaur hype a lot, but I'd say they were pretty well-known as the "badasses" of Earth's history before. JP built upon something that already had widely-spread cultural impact.
Which leaves The Matrix. And that is at least in my eyes
the film which really fits my requirements well and has tremendously more cultural impact than Avatar.