Over at Fivethirtyeight, Nate Silver argues that Donald Trump cannot win a war with FOX news. He has a good point. FOX is more powerful in the Republican echo chamber than most people can imagine. However, I think that this may be a war that The Trump doesn't need to "win".
Ezra Klein gives some background at VOX that you should read. After you've absorbed that, consider this: Trump is building himself as the leader of the conservative malcontents, the outsiders, the TeaBagger King, as it were. These people actually believe that FOX is too liberal. This is the group that Cliven Bundy and Sovereign Citizens everywhere belong to (as much as anarchists can belng to groups). They see FOX as part of the Republican establishment, part of the JEB! wing of the party. They need FOX, but they are naturally distrustful of it.
This is the group that Trump courts by taking on Megyn Kelly. He gets to stand up to the group that no other Republican has the balls to stand up to. Sure, in doing so he likely turned off around 30% of REpublican primary voters, but another 30% fell in even deeper love with him. I think that he comes out ahead on this in the long run.
Plus, since he is fighting with a woman and doing so on clearly sexist terms (attacking her for bleeding from her "wherever"), he bolsters his he man woman hater cred, which is a great way to win the dumb white male over 50 vote (you know, Republicans). This is win/win/win territory.
Seriously, in the battle for crazy and evil Trump is head and shoulders above the other Republicans.
On a side note, the audience during the debate cheered nearly everything Trump said. Only Trump excited the audience. When you're wondering who Republicans are and what they stand for, remember those cheers.