When Mike temporarily leaves after a fight with Gloria pertaining to her pregnancy, she gets distraught. So with Gloria on his lap, Archie tells her about the time when she was born.
Right before Mike and Gloria's wedding, Archie has a private father/daughter talk with her:
Archie: Do you remember when you was a little girl and you was a brownie? And you went up to the summer camp?
Gloria: Yes, but what does that have to to with anything?
Archie: Just lemme finish here. See, you thought you was gonna love the camp but it turned out you hated it.
Gloria: I called you up crying and begged to come home
Archie: I took the rest of the day off work, hopped on the next train, went up and brung you back. Cuz you was my little girl, and anything you wanted was ok by me. I just wanted you to know you're still my little girl, and you always will be.
Archie actually wanting to go to Lionel's engagement party, even telling Louise he was looking forward to it. Compare this you when the Jefferson's first moved in and he didn't want to step foot in a "colored" home for fear of being seen.
Archie's friendship with Lionel in general is this. While Archie often said offensive things unintentionally, he never did so maliciously and seemed to genuinely like Lionel. Lionel in turn never held Archie's attitudes against him. When Mike accused him of holding in anger and resentment against Archie for being a bigot, Lionel makes it clear this is not the case, and simply responds that Archie doesn't know any better.
Archie spends an entire episode dragging his feet about spending the weekend with Edith at the same hotel room they spent their honeymoon on their anniversary. Just when you think he might not even care about her at all, he sees her in her nightgown and is blown away.
Archie: Did... did you wear that dress on our honeymoon?
Edith: No, it's brand new.
Archie: Oh... for a minute there I thought... you look exactly the same as you did that night.
Afterwards, Edith thanks Archie for 25 great years of marriage. Archie is touched and replies:
Archie: Well, thank you too, Edith... after all, I couldn't have done it without you.
In the episode "Mike's Appendix", Archie pays for Mike's operation and asks Gloria not to tell him that he did. His reason for doing it: "He's the only Meathead you got."
When Stretch Cunningham dies, Archie attends his funeral and is surprised to find out that his old friend was Jewish. Naturally thus leads to some awkwardness on Archie's part, but he ultimately honors Stretch with a touching eulogy.
In the season 8 finale "The Stivics Go West", the final moments with Archie and Mike are this mixed with a Tear Jerker. Archie does everything he can to fight the fact the he's actually going to miss Mike but you can tell by the look on his face that when Mike tells Archie he loves him and will miss him that Archie feels the same way, he just can't spit it out.
In the season 9 finale "Too Good Edith", Archie chews Edith out for not telling him she had phlebitis so he could take care of her. He then climbs into bed with her and says this:
Archie: Oh Edith, you know somethin'? If the whole damn world was to go to the dogs, if I still had you standin' by my side, y'know? Or sittin' by my side, or lyin' by my side like this, everything would be just ok, y'know? I've been blowin' my own horn for a lot of years, but let me tell you somethin'; I ain't nothin' without you.
When The Jeffersons move on up, Edith goes to say goodbye to Louise. The exchange they have is absolutely beautiful
Edith: Did I ever tell you I love you?
Louise: Every moment we spent together.
In a Christmas episode, Edith's friend Beverly Lasalle (a famous drag queen Archie assumed to be gay and thus kept his distance from) is killed in a mugging. Archie is genuinely saddened, telling Gloria he wished he had told Beverly "what a nice fella she was."
Edith getting Archie to hug Mike when Joey is born.
In "Amelia's Divorce" Edith's boorish cousin-in-law makes a comment about Archie being "stuck" with Edith. Archie's response is so great, the audience applauds!
Archie: I ain't lookin' for no greener pastures 'cos Edith is green enough for me!
Archie is trapped in an elevator with a hysterical woman, an erudite African-American professor, and a Hispanic couple. When the wife goes into labor, Archie of course gripes and grouses the entire time. The range of emotions that play across his face is astonishing—from looking thoroughly annoyed and grossed out at the sound of the woman's screams, then genuinely worried when he asks why the baby isn't crying, then incredibly relieved when he finally does begin to cry. The husband gently invites him, "come and see my son", leading Archie to offer a fumbling congratulations on their little boy.
The episode "Cousin Liz", where Veronica talks (albeit with a lot of confusion and struggle) to Edith about her relationship with Edith's cousin Liz and about how it was "like a marriage". Edith figures out what she means and accepts their relationship.
The episode "Stephanie's Conversion". Edith finds out that Stephanie has been hiding the fact she is Jewish from Archie. When he finds out, he hits the roof but soon has a change of heart. At the end of the episode, he gives Stephanie a Star of David necklace and tells her: "You've got to love somebody to give them one of them (the necklace). I mean, you've got to love everything about them"
In one episode, Archie is busy with restaurant business to see Stephanie perform at a talent show. So Stephanie performs at the restaurant, making Archie regret missing it.
An early episode has Archie be friends with a former football player who is gay. It is charming that a show from the 70s has a homosexual come off as strong, charming, handsome and confident and even Archie sort of accepts him in his own way.
In a 20th anniversary special that aired in 1991, Sally Struthers, the actress who played Gloria, recalled that her Real Life father had died a few years before Family started, so she developed a special bond with Caroll O'Connor, who played her TV father. This makes certain scenes involving the two even more heartwarming, such as the first one listed above.
At the beginning of Part Two of "Edith's 50th Birthday"—the absolutely terrifying episode where Edith is almost raped—Archie finds out about the attack and immediately begins doing his best to care for Edith. He does whatever she asks him to do, and ultimately ends up just holding her. It's one of the best displays of his love for her in the whole series.
Archie: Yeah, see, but if I don't phone nobody important, or we don't drink nothing tasty, what're we supposed to do?
Edith: Hold me.
Archie: That's the best thing. I can do that.
In "The Unemployment Story, pt. 2", Archie helps talk a man out of committing suicide. He even strolls out on the ledge with the man, cool as a cucumber (until he realizes where he is), to show him a picture of Joey, and siting his grandson as something he has to live for. The man admits he also has a grandson, and Archie convince him not to jump since he wouldn't want the kid to grow up without his grandfather.
In the episode "The games Bunkers play" Lionel is challenged by Mike to say something to someone that he never had but always wanted to. He tells Edith how when his family moved in next door they were scared about moving into a white neighbourhood, but Edith made it clear that they'd always have at least one friend in the neighbourhood. Edith (and the audience) were very touched.