Here be weirdness.

For all his flaws as a father, Ging gave Gon some solid advice about apologizing to one’s friends. There are rules. It’s more than just saying “sorry.” You have to say what you did, and more importantly, make a promise: you promise them what you’ll do next time, and then you keep that promise, no matter what. 

It’s so important to Ging that Gon understands this, because even after telling him about Kaito’s ability, he tacks on: 謝り方間違えんなよ！！ 

It’s not just “don’t mess up your apology (or you'll be sent flying)," it’s “don’t mess up the way you apologize.” The action instead of the noun. 

On his way to Kaito, thinking about his apology, we learn that Gon is thinking about Killua too. He tells Leorio that he owes Killua an apology because he said some really terrible things to him when he got all messed up fighting the enemy. Gon is highly aware he hurt his friend. He is not oblivious. 

When Gon finally meets Kaito, we see him put his new knowledge of apologies into action. He says sorry, and when Kaito prompts, “What for?” Gon explicitly lays out what’s eating him up, “For not being able to fight alongside you, for not being strong enough. Next time, I’ll protect you!” He carries out the apology as best he can, even inspiring an eavesdropping ant to later own his mistakes to Kaito too.

The scene with Kaito is proof that Gon knows how to apologize to his friends, and that he is serious about making up for his faults. It’s existence is evidence that Gon apologized to Killua and made a promise to him too.

But why didn’t we get to see it? In my opinion, there are two potential reasons. 

Reason 1: Togashi wants to leave it to our imagination. 

It’s not uncommon for Japanese media to shy away from providing a perfectly polished emotional pay-off. Even in the romance genre, confessions are rarely explicit. The words “I love you!” might come out as, “The moon is beautiful tonight.” We know that Gon and Killua share an intense bond, and Togashi is well aware that people view their relationship through various lenses; to keep the apology and promise hidden, it allows each fan to insert what would make them happiest into the missing scene. It’s not a cop-out so much as a standard of Japanese storytelling. What can you read into the situation from what’s been implied? From the subtle? Between the lines? 

It’s true that there might be a "right" answer in Togashi’s mind, but if he’s chosen to be vague intentionally for the sake of keeping fans content, we’ll never know it. 

Unless!

Reason 2: Togashi will reveal the promise to us at a later date. 

It has happened multiple times throughout the series that an event occurred, but we didn’t learn the details of the event until later. Just as an example off the top of my head, when Gon is knocked out in the final round of the Hunter Exam, Satotz has to tell him what happened after he wakes up. Even in that re-telling of events, another seed of mystery is planted when Gon learns that Hisoka gave up his match after whispering something in Kurapika’s ear. We don’t find out what it was for several more episodes. Togashi has deliberately kept information from readers for the purpose of future storytelling, and this might be one of those moments.

It’s very possible we’ll discover what Gon’s promise was to Killua in the future, and I wouldn’t be surprised if we found out about it from a 3rd party, like Alluka. 

So, we know an apology was said and a promise made, but why did they have to separate? 

That’s tied to the promises they’ve already made that we do know about. 

Back in episode 38 of the anime, when leaving Whale Island, Gon tells Aunt Mito he’ll return after he’s met Ging. In episode 146, Killua tells Alluka he’s going to protect her from their family. As much as they want to stay together, and as much as we the viewers want them to stay together, it’s not the next right thing for them to do. It’s painful, but they need to spend some time apart. The day will come when they can be reunited. Alluka says it, Gon and Killua say it, and the lyrics of the song overlaying their parting say it:

いるよ 傍に 一番近く / I’m here, as close as I can be

今は ただそれだけでいいから / For now, that’s enough

いつかそっと 言いかけた / One day, you told me [about it] quietly,

夢の続きを 聞かせてよ / Let me hear the rest of that dream