J.K. Rowling addressed this in an interview in 2007, after the publication of the final book:
Did Hagrid ever get married and have children?
Oh, did Hagrid ever get married and have children? No. […]
Realistically, Hagrid's pool of potential girlfriends is extremely limited. Because with the giants killing each other off, the number of giantesses around is infinitesimal and he met one of the only [ones], and I'm afraid, she thought he was kind of cute, but she was a little more, how should I put it, sophisticated than Hagrid. So no, bless him.
The “only one” that Rowling refers to must be Maxime, so we can assume that any romantic feelings they had for each other weren’t enough to keep them together.
However, they didn’t part on bad terms. They clearly remained friends, as shown when they greeted each other at Dumbledore’s funeral:
Harry watched from a window as a gigantic and handsome olive- skinned, black-haired woman descended the carriage steps and threw herself into the waiting Hagrid’s arms.
— Half-Blood Prince, chapter 30 (The White Tomb)