Wiktionary
vb. To fight until a decisive and conclusive result is reached.
Wikipedia
Fight It Out is a 1920 American short Western film directed by Albert Russell and featuring Hoot Gibson.
Usage examples of "fight it out".
You're right and he's wrong, but it's more trouble than it's worth to fight it out.
Meanwhile the city is riven by internal strife as Senators Cicerius and Rittius fight it out for the post of Deputy Consul, and the criminal gangs The Society of Friends and The Brotherhood struggle for control of the dwa trade.
The bystanders interfered and tried to separate them, but they could not do it, and so allowed them to fight it out.
When we disagree with another person there are only three options: fight it out, withdraw, or negotiate a compromise (see method #10 in chapter 13 for resolving conflicts).
Best get him now, perhaps, and as many of the others as I can with the ring, then draw and fight it out with the rest.