I think it was mainly the Tennant persona that was in love with Rose.
I'm not sure. Honestly, I'm not.
Eccleston's Doctor didn't mind
flirting with Rose (see "The Empty Child"/"The Doctor Dances"), but he also tried to further her emotional relationships with others (bringing Adam and Jack onboard, the trip to Cardiff so she could hook up with Mickey in "Boom Town") and he was looking to replace her with Lynda on the GameStation.
Tennant's Doctor had
moments where there seemed to be more to their relationship than just flirting ("New Earth," "Tooth and Claw") and moments where he outright rejected her ("School Reunion," "The Girl in the Fireplace"), so there's certainly an ambivalence in the first half of season two about the Doctor's feelings toward Rose. I don't see a definitive shift toward romantic feelings on the Doctor's part in the latter half of the season. I
can see that Rose wants romantic feelings from the Doctor, and her behavior (and narration) in "Army of Ghosts"/"Doomsday" is predicated on her desires for a romantic relationship. The scene at Bad Wolf Bay
could indicate an interest, now unfulfillable, on the Doctor's part, and I've little doubt that was RTD's intention, but it's also ambiguous enough to support the contention that the Doctor simply wanted Rose to know that he made it safely out of Canary Wharf.
Personally, my read of that scene now is colored by season four, and I blame Rose's attempts to get back to the Doctor for the Daleks breaking reality, rescuing Davros, and stealing the Earth. (Before you say, "Buh?," here's the logic. The walls between the worlds were sealed. Rose and Pete-Torchwood's attempts to break through the walls weakened the fabric of reality enough that the Daleks were able to rescue Davros from the Time Lock, which enabled the Daleks to come up with their reality-busting plan, which culminated in the Daleks stealing Earth and the Doctor locking her irrevocably away in another universe. In other words, if Rose weren't selfish in her desires to be with the Doctor, we wouldn't have had "Journey's End.")
We
do know that the Tennant incarnation was incredibly fond of one companion, though, and it was a fondness that shaped his life beyond the norm -- Donna. Consider River's reaction to meeting Donna in "Silence in the Library." Then consider River's completely nonplussed reaction to meeting Amy. At some point in the Doctor's future (and, presumably,
not the Tennant incarnation), he and River had a conversation about Donna, and to judge by River's reaction to Donna, it was likely a Very Big Thing. The interstitial scene between "Flesh and Stone" and "Vampires of Venice" on the fifth season box set (the Doctor's reaction to using the word "mate") is indicative of this as well. I'm not suggesting that the Doctor was in love with Donna, but Donna was clearly very important to the Doctor, perhaps, even likely, more than Rose and Martha were.