The short answer is: Maybe. The evidence we have is contradictory.
the Doctor said in the TV movie that he was half-human on his mother's side, the Master believed him, and he proved it by opening the Eye (which was keyed to only open for half-humans, so it wouldn't work for the Master).
However, in Journey's End, the DoctorDonna said she was special because she had a little bit of human, while the Doctor was just a time lord . And she knew everything the Doctor knew at that point, due to the meta-crisis, so she'd know. The fact that the Woman in The End of Time appeared to be a Time Lady and was at least possibly his mother seems to back that up.
An ongoing theme since Steven Moffat took over the series is "the Doctor lies". Although this has yet to directly address the statement made in the TV movie, it does offer a potential out for getting around this issue - the Doctor may have simply lied to Grace. The Master independently makes this statement in another scene (giving a possible reason for the Doctor lying, if he was somehow trying to deceive the Master). While this doesn't explain why the Eye opened for the Doctor and not the Master, it's worth noting that the Master never actually checked; he just assumed the Eye wouldn't open for him, so the whole thing could conceivably have been a bluff.
On the other hand, another ongoing theme of the Moffat era is "History can change". Again, this has yet to be used to address the half-human statement, but again it offers a potential out. For example, the Doctor's explanation of Amy's conflicting histories in Good Night could apply to his own conflicting histories: just as Amy has memories of never having had parents alongside memories of growing up with her parents, the Doctor could have memories of a human mother alongside memories of a Gallifreyan mother. It's not a question of which one is true, just which one is true at any given moment, and even that isn't usually all that important.
Adding to the confusion is the introduction in Human Nature of the chameleon arch, a device capable of physically changing the Doctor (and, later, the Master) into a human. The Doctor could have made himself half-human and rekeyed the Eye specifically to protect it from the Master, or from other Time Lords.
Some speculation has also centered around the situation the Doctor (and Grace) were in at the time, being questioned by a security guard and trying to confuse the issue of the Doctor being an actual 'alien'
The other media make things even more complicated.
The New Adventures (NA) novels, which followed the classic series, showed that the Doctor, and all Time Lords, are born out of giant machines called Looms. While some of the novels say that there are at least legends of "womb-borns", Time Lords bred naturally in secret by those who have found ways around the Curse, the Doctor is explicitly shown to be part of a Loomed family. However, Lungbarrow (deliberately written to keep the NAs and the TV movie out of contradiction) established that the Doctor may be the reincarnation of the Other, who may be the son of Andred and Leela, which would mean he was (in a convoluted way) half-human despite being Loomed.
The Eighth Doctor Adventures (EDA) novels that followed the movie featured major storylines where the Doctor's history and Gallifrey's were edited repeatedly, changing his origins radically. At one point he even had 17 origins (including being a 49th century human scientist) that were all true and all contradictory. At the end of the series, his father was a Gallifreyan and his mother was a human (probably Ulysses and Penelope Gate). But there's no reason to believe that's any more the "final answer" to the Doctor's origins than any of the earlier ones, since the novel leaves the Doctor in mid-stream, heading toward another history-changing event (the re-creation of Gallifrey).
Finally, in the IDW comic The Forgotten, the Doctor says that he once pretended to be half-human in order to fool an enemy. It's a bit vague in-story, and it happens in an unreal world, as part of a conversation where the Doctor is testing "Martha" to find out who she really is. However, the editor confirmed that this is intended to say unequivocally that the Doctor is not half-human, and in the TV movie he was either bluffing or using a complicated ruse (possibly with a chameleon arch, as explained above).
There are many ways to explain away these contradictions. The Doctor could be lying in The Forgotten (the editor's out-of-universe statement to the contrary). There could easily have been another history-changing event between the last EDA and the IDW comics (for example, the Last Great Time War and destruction of Gallifrey, which echo the Second War in Heaven and destruction of Gallifrey in the EDAs, which we saw change the Doctor's origins). One EDA author, Larry Miles, argues that the EDAs are canonical, but concern an alternate history. Meanwhile, the BFAs establish that non-Gallifreyans can become Time Lords, so the Woman could be a Time Lady and the Doctor's mother and still be human. And so on.
None of these explanations have been established in-universe (although, as mentioned above, some of them at least have parallels that have been).