Probably. But these are murky waters, constitutionally. On the face of it there’s no way Brexit can happen without the approval of Parliament (’nothing can be done which Parliament may not undo’) since Parliament is sovereign. But there are some who argue that there are getarounds. There’ll certainly be an almighty row if Johnson really does try to bypass Parliament completely - there’ll be accusations of a coup, etc. Effectively it would mean that direct democracy (the referendum) replaces representative democracy in constitutional authority.
While the Speaker’s powers are extensive, they’re largely procedural. However, the current speaker is a maverick with a high opinion of himself and his office, and he’s already broken with precedent on more than one occasion. What he can’t do is force the government to do anything, or prevent the government from doing something, outside Parliament.