...And as I've also noted in an earlier thread: It's easy to read this as Kirk not falling in love with anybody at all, but merely executing a clever campaign to defeat his foe. Just like he always does with the men and women he meets.
Spock and McCoy really accomplish nothing here. Their lines of inquiry into the mystery of why Flint isn't helping them, as well as their attempts to overcome Flint's obstructionism, are seemingly logical but miss the mark completely. Kirk in turn cuts right to the chase, understanding what makes Flint tick and hitting him in the soft (Flint, unlike M-4, is quite defeatable in fisticuffs, and Rayna can be swept off her feet with basic techniques Kirk is well versed in) and sensitives (Rayna is a major weakness in Flint's armor).
In the end, Kirk weeps. But not for Rayna, at least not by his own words. He's weeping for Flint, for whom he apparently didn't feel erotic love, but whose fate is a sorry one, and in many ways a painfully close match of Kirk's own lot in life.
No, you can't read that at all based on anything in the episode.
KIRK: Stay out of this. We're fighting over a woman.
SPOCK: No, you're not, for she is not.
(Flint thumps Kirk, then repeatedly throws him across the room)
RAYNA: I cannot be the cause of this. I will not be the cause of this. Please stop. Stop! I choose where I want to go.
(The men stop fighting in astonishment)
RAYNA: what I want to do. I choose. I choose.
RAYNA: No. Do not order me. No one can order me!
KIRK: She's human. Down to the last blood cell, she's human. Down to the last thought, hope, aspiration, emotion, she’s human. The human spirit is free. You have no power of ownership. She's free to do as she wishes.
SPOCK: Gentlemen, I urge you to stop. There is a danger.
FLINT: No man beats me.
KIRK: I don't want to beat you. This is no test of power. Rayna belongs to herself and she claims the human right of choice to be as she wills, to do as she wills, to think as she wills.
FLINT: That's what I've worked for.
KIRK: Rayna, come with me.
RAYNA: I was not human. Now I love. I love.
(She collapses. McCoy checks for a pulse)
FLINT: You can't die.
KIRK: What happened?
SPOCK: She loved you, Captain. And you, too, Mister Flint, as a mentor, even as a father.....
.....MCCOY: Well, I guess that's all. I can tell Jim later or you can. Considering his opponent's longevity, truly an eternal triangle. You wouldn't understand that, would you, Spock? You see, I feel sorrier for you than I do for him because you'll never know the things that love can drive a man to. The ecstasies, the miseries, the broken rules, the desperate chances, the glorious failures, the glorious victories. All of these things you'll never know simply because the word love isn't written into your book. Goodnight, Spock.
SPOCK: Goodnight, Doctor.
MCCOY: I do wish he could forget her.
(McCoy leaves. Spock goes over to Kirk and initiates a mind meld)