@Verus:
Every, and I mean every respectable goverment would deny any guilt regarding the incident and drop it all on Fujitora's. Probably give him dishonorable discharge as well.
I disagree emphatically with your interpretation of what is going on, seemingly on pretty much every level.
First, when Tashigi arrested Crocodile, she said, "Navy Headquarters under direct authorization from the World Government revokes your license to seize enemy ships and strips you of all rank and authority forever". Which suggests to me that up until they had received that, Smoker was exceeding his legal authority by opposing a Shichibukai, but he was doing it in the name of Moral Justice (aka Doing The Right Thing) and he figured that he would uncover evidence in the course of his investigation to prove himself later. His tendency to do exactly that is why he's seen as such a loose cannon and combined with his insubordinate attitude, is often in trouble with his higher ups (as seen from his conversations with Hina, for example).
Here on Dressrosa, Fujitora was sent to pick up one rogue Shichibukai, and to investigate another one on behalf of the Marines (and World Government). He was given information that both of them would be meeting up on a certain beach on Green Bit at 3pm, and then, at like 2:58, was called by the Fleet Admiral to inform him that the report of Doflamingo's resignation was incorrect. Fujitora was even like 'What do you mean, the Marines got misinformed?', so it's pretty clear he's wondering what the hell is going on.
Then he sees Doflamingo getting called "Joker" and just stands there looking disapproving as he hears Doflamingo pretending to have abdicated. He knows something's up, but he's wondering how the hell Doflamingo managed to pull something like this on the Marines. He continues to listen to Law and Doflamingo's conversation, so he's got to be wondering the same thing Law is at this point. Then, when he suggests Doflamingo has been up to some stuff that might violate the rules of being a Shichibukai, Doflamingo says he'd better have 'a heap of evidence'. He had to suspect from then that Doflamingo was well-connected and could pull the strings of the World Government.
We already saw him talking about counting the number of people who need protecting, so this was pretty clearly a priority of his. Once Law had been subdued, however, Doflamingo took the time to explain that he came from a family of Celestial Dragons, which revealed more of the situation to Fujitora, although he says that he is no longer one, but doesn't tell the full story. At this point, however, Fujitora's mission is clear: apprehend Law and the Strawhats. Still, he clearly doesn't trust Doflamingo, and Doflamingo is obviously aware of his suspicions.
After the Swirlystraws escape, he heads off only to hear that the fight has returned to Dressrosa and arrives only to find Doflamingo shooting and capturing Law in the middle of Dressrosa, so they go off to have a chat before Fujitora decides how to proceed. While talking, Doflamingo points out that he is not only a Shichibukai, but the legitimate king of Dressrosa, and the Strawhats are rampaging around his kingdom, and shouldn't the Marines be doing something about this? Fujitora agrees, but explains that he's not Doflamingo's ally, he's just trying to minimize the damage done in the fight, which is otherwise not the Marines' business. He then explains that he plans to deal with Doflamingo eventually - by eliminating the Shichibukai.
Essentially, he's declaring that he's going to do this by the book. And by the book, Doflamingo is a legitimate king of a World Government nation and a Shichibukai, and what he says is law on the island. Fujitora is going to let him do whatever he wants to, and the Marines will stand by and protect the populace as best they can. By the book, he can't arrest Doflamingo without authorization from the World Government, and with the Birdcage up, he can't obtain that authorization. And since he doesn't know the full extent of Doflamingo's connections, that also kind of helps cover his ass.
Obviously, longtime Marines like Maynard are morally outraged by this. Sure, it's not strictly by the book, but they can prevent harm to the citizens if they take down Doflamingo now, and surely they'll be exonerated later. Fujitora, however, is going to be a stickler for the letter of the law in order to allow the situation to become worse and worse, and the only thing he'll do is try to protect citizens until and unless he has no choice but to act. For now, though, he's hoping that he can minimize casualties by taking out Law (again) and the Strawhats.
However, once he sees the army of allies joining Luffy, and Sabo the Revolutionary (and new user of the mera mera no mi) shows up to block the way, he realizes that there was even more to the situation than he had realized, and he makes the decision to let Luffy try to beat Doflamingo; in short, he bets on Luffy (the 'same bet' that Sabo is making). That way, when Luffy beats Doflamingo, Fujitora was protecting the citizens and obeying the letter of the law, and Doflamingo was out of control and immune to legal repercussions. Yes, it was an outrage, but Fujitora didn't feel he was allowed to violate the law in order to enforce his morals, especially when it would mean overthrowing the accepted king of a nation.
You think the various monarchies that seem to make up most of the World Government aren't going to accept that as an excuse from the Marines? When it's basically reassurance that the Marines will support them, no matter what they do?
TL;DR Fujitora's actual authority here is highly questionable; he knows Doflamingo is extremely well-connected; and Maynard objects on moral grounds which likely technically exceed their legal authority