"So exactly how do we justify this movie as a "happy" ending?"
I don't see their relationship as being romantic. I see him as being like an uncle for her -- or possibly a teacher, or employer.
The idea of the 2 of them getting married, makes me want to violently barf -- and then afterwards violently barf, again.
I too don't see the ending as romantic in the least. I do believe Eliza develops a sort of infatuation for the man who plucked her out of the gutter [as cruel as he was and is.] but by the end of the movie [and book/play], Eliza is supposed to have grown out of it and is left with a bittersweet affection.
As for Henry Higgins, he is depicted in all versions as having a bit of a mother-complex, in that all woman are forever vastly inferior to his mother. He has never and will never be physically interested in a female, let alone romantically and that suits him just fine. His affection for Eliza is one of familiar comfort and that of an egotistical creator who wishes to possess and admire his work for purely selfish gratification, forgoing the feelings of said 'creation'.
The fact that this went over to the head of so many people from the start is baffling to me, because I saw the complexity clear as day, but apparently not many others did or have.
G.B. Shaw was left no choice but to write a sequel of sorts and tack it on to the end of the published work a few years after it's release. He explains why such a thing as a romantic or living relationship between Eliza and Higgins would never work and should never be and then says what happens to the characters after the events of the play in a couple of short sentences.
He ends with, "Eliza is still a part of Wimpole Street and she is still interested vaguely in Higgins, but she keeps him at a distance and holds his derisions of Freddy to a minimum. She is also very much beloved by Colonel Pickering, and she returns his love. In Shaw's words, Eliza "likes Freddy and she likes the Colonel; and she does not like Higgins and Mr. Doolittle. Galatea never does quite like Pygmalion: his relation to her is too godlike to be altogether agreeable."
[basically, if she loves anyone she loves colonel pickering, so stick that in ur pipe and smoke it. :b]