This is a difficult question to answer. Their relationship is platonic, but as his companion she's as close to him as anyone ever gets. They're kind of like really good friends.
- It goes deeper than that. Rose outright says she loves the Doctor in "Doomsday" and not in a "you're a very close friend" sort of manner. The Doctor himself cannot respond (for reasons explained in "School Reunion"). But if you watch the episodes with Rose, she clearly acts as his girlfriend, even if he doesn't (intentionally) reciprocate. The only thing missing - we assume, as it's never shown on screen - is the snogging (kissing the clone in "Journey's End" doesn't count.