Does Spock fall in love with Nurse Chapel?

Did Nurse Chapel and Spock have a relationship in the original series?

Was it mentioned?

Or were they just friends who happened to be in the same place at the same time? Or did he only care for her as a friend until she died and then he was heartbroken? Or did he really love her but couldn't admit it in a way that made him not very likeable in a later canon?
I suspect it's the latter. Nurse Chapel and Spock weren't involved romantically, though she loved him romantically. She was in love with him like Spock is with Uhura, but it wasn't sexual. After Spock was in the hospital, I think she tried to get over him and move on, but his death hit her hard. It hurt her, just as it would have hurt any other girl. She was probably in a very depressed state afterwards, but she didn't give a damn about that. All she wanted to do was to find out how her friend died, and then to die herself. Because she already knew she was going to die anyway, being a Vulcan didn't mean that she didn't know she was going to die. What good did it do her to know that, if she was going to die anyway? She found out what had happened to Spock, and that she could die if she didn't get that cure, so she went to the closest medical facility and was injected. And then, she just lay down and waited for death.

When Spock later met Chapel again, it was because Kirk decided that there was something going on between them and ordered them to go back together and look for the cure. Spock had the cure first, but it didn't work for him. He came back to Kirk with Chapel and told him that she was waiting for the cure and also said that she was waiting for him to come back. If Spock had come back at this point, she would have died with some dignity. Kirk let her know that Spock was sorry for dying. There's no evidence that she had any feeling about it. Even when he got cured, there's no evidence that she ever talked to him about it. For all we know, she might have said "Sorry for what?" or "You're lucky to have a friend like that." but I seriously doubt that. The only thing that we know for sure is that when they were together, Spock never admitted that he loved her. That's the one thing that Spock will never be forgiven for.

Why did Spock kiss Nurse Chapel?

There has been much ado about whether Kirk or Spock should win the affections of Nurse Chapel, who is played by Majel Barrett.

Although the matter is settled with the two captains kissing their respective officers and letting loose some of the steam they have accumulated over the course of the series, it was never quite as clear-cut as that. So why did they kiss, who were their motives for doing so and why were they kissing one another?

Kirk and Spock are already in love with one another; the entire premise of the Kirk/Spock relationship is based on that premise. After the end of episode A Private Little War we see that the rest of the crew loves Kirk as well, which could also be regarded as the premise. And there we also have the problem that Captain Kirk would never dream of asking one of the crew for a date, he would never think of anything like that. But let's look at it from a logical Spock viewpoint: When he left for his assignment in space he gave her his handkerchief, then he went off into space on a perilous mission to discover the Romulans and their secrets. Of course he wants to remain the good and loyal servant of Captain Kirk.

But what would he do if he had the opportunity to date a member of the crew? How would this come about? Obviously she would come along too, when Spock is assigned to the spaceship Enterprise. That seems like a logical step, although it takes the ship back to the planet San Francisco which was the setting of their first episode together. In this new environment Spock would find it difficult to go on dates. In San Francisco the crew does not know about his Vulcan origins, Spock will not reveal anything about his past and so we will probably never learn exactly where he is hiding out.

If Nurse Chapel goes along anyway, well, it might because she is attracted to Spock but she never lets him know that and for him the situation seems strange, to say the least. He cannot figure out what it is all about and he keeps having to keep telling himself that he must not become infatuated with her. His emotions are simply not strong enough for such an emotion at this time. And Spock says no so firmly that you cannot accuse him of being cowardly.

That's why the whole matter of Kirk and Spock kissing comes to a head.