First of all, I'm very unsurprised that you can get special and general relativity out of something like this. Relativity fundamentally just isn't that complicated and you can see what are basically relativistic phenomenon pop out of all sorts of natural setups where you have some sort of space with an emergent distance metric.
The real question is how this approach handles quantum mechanics. The fact that causal graph updates produce branching structure that's consistent with quantum mechanics is nice—and certainly suggestive that graphs could form a nice underlying substrate for quantum field theory (which isn't really new; I would have told you that before reading this)—but it's not a solution in and of itself. And again what the article calls “branchial space” does look vaguely like what you want out of Hilbert space on top of an underlying graph substrate. And it's certainly nice that it connects entanglement to distance, but again that was already theorized to be true in ER = EPR. Beyond that, though, it doesn't seem to really have all that much additional content—the best steelman I can give is that it's saying “hey, graphs could be a really good underlying substrate for QFT,” which I agree with, but isn't really all that new, and leaves the bulk of the work still undone.
That being said—credit where credit is due—I think this is in fact working on what is imo the “right problem” to be working on if you want to find an actual theory of everything. And it's certainly nice to have more of the math worked out for quantum mechanics on top of graphs. But beyond that I don't think this really amounts to much yet other than being pointed in the right direction (which does make it promising in terms of potentially producing real results eventually, even if doesn't have them right now).
TL;DR: This looks fairly pointed in the right direction to me but not really all that novel.
EDIT 1: If you're interested in some of the existing work on quantum mechanics on top of graphs, Sean Carroll wrote up a pretty accessible explanation of how that could work in this 2016 post (which also does a good job of summarizing what is basically my view on the subject).
EDIT 2: It looks like Scott Aaronson has a proof that a previous version of Wolfram's graph stuff is incompatible with quantum mechanics—if you really want to figure out how legit this stuff is I'd probably recommend taking a look at that and determining whether it still applies to this version.
Thanks for writing this. I hesitated before commenting, because I am not an expert on physics, but something just felt wrong. It took some time to pinpoint the source of wrongness, but now it seems to me that the author is (I assume unknowingly) playing the following game:
1) Find something that is Turing-complete
The important thing is that it should be something simple, where the Turing-completeness comes as a surprise. A programming language would be bad. Turing machine would be great a few decades ago, but is bad now. A system for replacing structures in a directed graph... yeah, this type of thing. Until people get used to it; then you would need a fresh example.
2) Argue that you could build a universe using this thing
Yes, technically true. If something is Turing-complete, you can use it to implement anything that can be implemented on a computer. Therefore, assuming that a universe could be simulated on a hypothetical computer, it could also be simulated using that thing.
But the fact that many different things can be Turing-complete, means that their technical details are irrelevant (beyond the fact they they cause the Turing-completeness) for the simulated object. Just because a Turing machine with 1-dimensional tape could simulate the universe, it doesn't mean that the universe is "truly 1-dimensional", "truly consists of discrete cells containing symbols", or anything like that. It just means it could be simulated on such system, because such system is Turing-complete; nothing more, nothing less.
So it's simultaneously "yes, you could simulate A using B" and "A and B are quite different". However, you can...
3) Find some shallow analogies between the thing and the laws of physics
Remember, these are analogies, so you can point out any similarity you find, and change the topic whenever the similarity is exhausted. For example, the Turing machine has a tape consisting of discrete cells... and if you squint, that is kinda like the quantum physics, because the quantum physics also has some discrete values, and... uhm, end of paragraph. But also the Turing machine moves at a limited speed, at most 1 step per turn... and that is kinda like theory of relativity, you know, with the limited speed of light. And perhaps the tape looks like a string, which would point towards string theory, who knows. You would notice different similarities with the graph replacement rules, but you could find some if you try hard enough.
This gives the fake impression that we have found a similarity deeper than "Turing-complete, therefore universal, therefore allows to simulate any universe". (If you can simulate any universe, it means you cannot conclude anything useful about the universe from the fact that it can be simulated.) It gives the impression that this specific system is somehow more relevant to our universe than any other Turing-complete system. That it provides insights into the true nature of our universe. Which in fact it does not, because all we have is the Turing-completeness and a few shallow analogies. The fact that X can simulate a universe, and also has a few shallow analogies with the known laws of physics, means less than it may seem.
(So, thanks for confirmation that the analogies with laws of physics are indeed gibberish.)