Why does Eliza come back to Henry Higgins without getting any apology?

As with any teacher-student relationship, it's best if there's a firm break between being in that relationship and being in a relationship of equals.

That's what this scene is. Eliza is now fluent in upper-crust English. She can "Stand on [her] own without [him]." So, what she's saying is that she doesn't need him as a teacher any more. And, based on his reaction, we are led to hope that he no longer sees her as a student.

In the time between then and the end of the film, when she's hanging out with Freddy, she realizes that, while she may not need Henry as a teacher any more, she needs him as a man - she needs a partner. She's fallen in love with him. And, based on I've Grown Accustomed to Her Face, he has similarly fallen in love with her.

Freddy is offering to love her but not in the way she wants - his love is childish, it's all about making pretty speeches. This is pretty well discussed in Show Me. She wants someone who does things that show he cares rather than flowery language. Henry doesn't do flowery language but he's done everything else for her. He's given her a better life, made her self-sufficient.

So, when she returns to him, she returns as a woman who loves a man, not as his student and he is a man who loves her, in return.

As to why she doesn't require an apology - first, I don't think she'd expect one from him. That's not the sort of man he is and she's already said that she's tired of words. Secondly, I'd say she actually gets one, of a sort. There's an action in that scene that more than makes up his "apology". He enters the room and plays a record of her voice from when she first arrived. This confirms to her that he loves and misses her in a way that's much better than any spoken apology ever could.

Have you read Shaw's explanation of how the story ends in Pygmalion?

The rest of the story need not be shown in action, and indeed, would hardly need telling if our imaginations were not so enfeebled by their lazy dependence on the ready-makes and reach-me-downs of the ragshop in which Romance keeps its stock of "happy endings" to misfit all stories. Now, the history of Eliza Doolittle, though called a romance because of the transfiguration it records seems exceedingly improbable, is common enough. Such transfigurations have been achieved by hundreds of resolutely ambitious young women since Nell Gwynne set them the example by playing queens and fascinating kings in the theatre in which she began by selling oranges. Nevertheless, people in all directions have assumed, for no other reason than that she became the heroine of a romance, that she must have married the hero of it. This is unbearable, not only because her little drama, if acted on such a thoughtless assumption, must be spoiled, but because the true sequel is patent to anyone with a sense of human nature in general, and of feminine instinct in particular.

Eliza, in telling Higgins she would not marry him if he asked her, was not coquetting: she was announcing a well-considered decision. When a bachelor interests, and dominates, and teaches, and becomes important to a spinster, as Higgins with Eliza, she always, if she has character enough to be capable of it, considers very seriously indeed whether she will play for becoming that bachelor's wife, especially if he is so little interested in marriage that a determined and devoted woman might capture him if she set herself resolutely to do it. Her decision will depend a good deal on whether she is really free to choose; and that, again, will depend on her age and income. If she is at the end of her youth, and has no security for her livelihood, she will marry him because she must marry anybody who will provide for her. But at Eliza's age a good-looking girl does not feel that pressure; she feels free to pick and choose. She is therefore guided by her instinct in the matter. Eliza's instinct tells her not to marry Higgins. It does not tell her to give him up. It is not in the slightest doubt as to his remaining one of the strongest personal interests in her life. It would be very sorely strained if there was another woman likely to supplant her with him. But as she feels sure of him on that last point, she has no doubt at all as to her course, and would not have any, even if the difference of twenty years in age, which seems so great to youth, did not exist between them.

(continues)