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.
I've tried to read through the linked page, and swapped to `academic reading' (checking the pictures, and sometimes the first and last line of paragraphs) halfway through. I think this is not viable.
There is a host of "theories of the universe" with a similar structure on a meta-level, consisting of some kind of emergent complexity. It is important to keep in mind the strength of a theory lies in what it forbids, not in what it permits. To date most theories of the universe fail this test hard, by being so vague and nonspecific that any scientific concept can be pattern-matched to some aspect of it. Judging by what I've read so far this is no exception (and in fact, I suspect that the reason Wolfram references so many big scientific theories is because large concepts are easier to pattern-match, whereas specific predictions are not as open to interpretation). Why will his patterns produce Einstein's equations (note that they currently do no such thing, he states we first need to "find the right universe"), and not Newton's, or Einstein's with double the speed of light?
As always with these nonspecific `theories' it is very difficult to nail down one specific weakness. But currently all I'm seeing are red flags. I predict serious media attention and possibly some relevant discoveries in physics (some of the paragraphs sounded better than all other crackpot theories I've seen), but the majority of it seems wrong/worthless.
So if BQP turns out to be larger than P, will Wolfram announce his theory wrong or rather modify its predictions? This would be a great test of... himself.