A complete accounting has not been made, and the show itself makes it difficult to tell because it rarely has characters come out and say "I love you" to the Doctor. It has also from time to time suggested in retrospect that characters have been in love with the Doctor when the original episodes were ambiguous.
Speaking for TV alone, the following female characters are confirmed to have "fallen for" the Doctor in some fashion:
Cameca ("The Aztecs") - fell for the First Doctor; the only known example of the Doctor experiencing anything approaching romance during the original series.
Grace Holloway ("Doctor Who" - the TV movie)
Rose Tyler ("Doomsday")
Sarah Jane Smith (as revealed in "School Reunion")
Martha Jones ("Utopia")
Joan Redfern ("Human Nature"/"The Family of Blood")
Astrid Peth ("Voyage of the Damned")
River Song (fourth wife of the Doctor. Multiple episodes, though it really wasn't until "The Husbands of River Song" that this was discussed at length on screen)
Clara Oswald (per "The Time of the Doctor", and according to Jenna Coleman her statement of "I love you" in "Mummy on the Orient Express" was directed at the Doctor. The prequel minisode "She Said, He Said" has her discussing the difficulty of not falling in love with the Doctor, as well)
Queen Elizabeth I ("The Day of the Doctor", in which she marries Tenth Doctor)
Amy Pond is not included in the above list because, despite her attempt to seduce the Doctor in "Flesh and Stone", and other flirting, the series married her off to Rory and established her as the Doctor's mother in law, imposing a familial love rather than romantic love in her case.
Missy (aka The Master) referred to the Doctor as her boyfriend in "Deep Breath" and has kissed and flirted with the Doctor romantically in episodes like "Dark Water", however given the nature of the character it's impossible to tell if any of this reflects genuine feeling. Similarly, while Tasha Lem is clearly attracted to the Doctor in "The Time of the Doctor" and it's hinted that there may have been a past relationship, the episode doesn't go into enough detail to tell.
Although the Fourth Doctor and Romana were shown in romantic circumstances a few times (most notably in "City of Death"), and in real life the actors were dating and later married, the franchise has never openly suggested that Romana was ever in love with the Doctor.
We are aware of at least two additional marriages: Marilyn Monroe ("A Christmas Carol") and apparently Cleopatra ("The Husbands of River Song"). Assuming the Doctor's offspring who begat his granddaughter Susan was not related to these two, or River Song and Queen Elizabeth I, there is at least one other individual with romantic feelings towards the Doctor to the point of bearing at least one child to him. (Note, however, that some of the novels, most notably "Lungbarrow" (novel), postulate that Time Lords do not reproduce in the same way as humans.)
Audio dramas and novels have also either implied or directly indicated other female characters having fallen in love with the Doctor. One notable example is Benny Summerfield who actually goes to bed with the Doctor in "The Dying Days" (novel)) (confirmed in the Big Finish audio drama "Benny's Story" (audio story)).