I think the ending is the most precious, touching part of My fair lady. Not because they end together [sorry for the spoiler, but it was just said above, haha] and it's romantic — on the contrary, it's just because it is not sweet and romantic. I've grown accustomed to her face has brilliant lyrics that express exactly how their relationship, Eliza and Higgins', go. Eliza could sing Without you forever, but it's more than clear she can't do without him, and neither can him. He is, like the lyrics go, second nature to her now, like breathing out and breathing in. They are, then, part of each other's life, a habit one can always break and yet they've grown accustomed to each other's face, they fell in love without noticing it. It's exactly the opposite of a romantic love, which is torrid, incontrollable and devastating. It's just a statement of love, of something they've been feeling without knowing — and even denying it.
The last line of the movie just confirms it.
"Where the devil are my slippers?"