How do you think Barnes would react to nurse who caught his eye already having a relationship with someone like Gator, Chris or maybe even bunny?
You know what war produces plenty of? Widows.
Abandoned significant others.
Potentially broken up families.
The person of interest doesn't even have to be a nurse; could be a civilian. Could be someone said soldier(s) meet on R&R, could be a spouse, hey, could even be a sweetheart from a picture tucked into someone's breast pocket or locket --- could be a pen pal they've never even met; Barnes wants what he wants and the individual on the receiving end of your affection might just also mysteriously find themselves on the receiving end of a whole lot of tasks given as a direct order by the Staff Sergeant and said tasks might just be high risk or downright dangerous, as a mother of all coincidences. Can't refuse a direct order. Not without a court martial looming over your head for disobedience --- so Barnes is here deliberately weeding out the opposition by doing what he simply does every day; dishing out authority, notwithstanding the fact that I envision the man as incredibly jaded, cynical and bitter. Someone else in love just makes him roll his eyes. Say, someone like Gator, Chris or Bunny's in a happy relationship? They make a show of it? Or worse yet, they're secretive about it to the degree Barnes notices the conspiratorial behavior? Unrealistic! Waste of time! A folly of the childish and deluded! It's almost like the man wants to punish them for having their heads ''in the clouds'' or being idealistic over anything at all. For having something distracting them from the business of war.
For making them soft, the way he'd see it.
-"Y'all focus on the here and now!"- He might announce openly, as a memo.
-"Crap from the World belongs back in the World, not out here where there's shootin'. Gettin' your balls blown clean off over a letter."-
(In present day jargon, put away the damn phones, kids and pay attention to the homework).
Worst is, Barnes entirely means those words. He believes in them.
Sees love as a distraction all while simultaneously coveting it.
And hating the fact he does.
The soldier Barnes has it in for might not necessarily die per se because he'd view that as a waste of troops and I doubt he exactly likes that (in fact, he was shown to not like that) but they'll certainly go through so much trauma and unease Barnes hopes to god it cures them of their need for anything from the outside world, including you. By the time he's done with them, and by the time this war's done with them, they'll break it off with you on their own or simply disappear from your life; in modern terminology, go ghost on you --- a feeling Barnes very well understands, having been in The 'Nam for how many years now? Irony of all ironies, Barnes conducts said series of punishment and hardass behavior by effectively breaking up the relationship all while hiding his efforts under the facade of military strictness. And once the targeted soldier is so jaded themselves they've no interest left in love? The task is complete. Barnes might not make a move on the abandoned significant other himself immediately, but the fact that the path to them is open? That whatever rival he had is gone? That he achieved this? He already sees them as his in so many ways, it's just that they, the soldier they've broken up with and nobody else knows it yet. Barnes is a general destroyer of happiness and creator of misery; messes up someone else's joy and then goes and claims said joy for himself all while putting up a general front how he's 'above that shit'. He isn't, in fact, above that shit. He'd outright kill for that shit and if push came to shove and disciplinary measures and burdening the particular private he's made a covert enemy of doesn't cut it, flat out fragging will.