Doctor Strange initially discovers the ability and perhaps would have attempted to heal his hands if he was not interrupted.
After he was interrupted, he did not have another opportunity because of the fast paced action afterwards.
By the end of the movie, his focus is no longer on healing himself and he may view his hands of a reminder of how he got where he is.
Also possible is that Doctor Strange believed Wong.
Wong told him there were serious consequences for using it, and Doctor Strange apologized and only used it later in the movie for important matters such as saving the world.
scifi.stackexchange.com/users/70537/Goose
Wong tells Strange about the dangers of using the Eye, including the possibility of time loops and paradoxes.
There is no evidence in the movie that the Eye can be used to reverse time for a single object, so using it to fix his hands would require reversing time all the way back to prior to the accident.
This would create a paradox, since he would no longer have a reason to end up searching for Kamar-Taj and a way to heal his hands.
Of course, he does use the Eye to reverse time later, and even intentionally create time-loops, but only when there is no other recourse, and no real danger of creating a paradox.
scifi.stackexchange.com/users/5184/phantom42
On several occasions during the film it is pointed out that using the Eye of Agamotto to manipulate time is, at best, risky.
It’s not hard to imagine that turning back time for just one part of his own body might cause much more damage than Strange’s original injury, to him and/or the space-time continuum at large.
scifi.stackexchange.com/users/440/Paul D. Waite
He chose not to.
He got past his ego and finally realised that it was not about him, as the ancient one said.
Healing his hands would mean he is choosing to live as the old Strange and not embrace his responsibilities as a sorcerer.
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It's because he doesn't care to heal his hands.
As he descovered his new calling.
He wasn't obsessed with his past profession and life anymore
scifi.stackexchange.com/users/90874/Matthew Laloggia
In the movie,     Strange finds the Eye of Agamotto and notices it can manipulate time. He can manipulate individual objects such as an apple from being fresh to completely moldy and rotten. It is later shown he can use the time warping power with one hand and that there is at least one Master who is missing a hand altogether and can still perform magic spells.    Why didn't Strange use this to heal his broken hands, one at a time?
Why didn't Dr. Strange do this to heal his hands?
scifi.stackexchange.com
<marvel-cinematic-universe><marvel><doctor-strange><doctor-strange-2016>
scifi.stackexchange.com/questions/144584
scifi.stackexchange.com/users/28010/The Unknown Dev
There are many possible explanations for why Doctor Strange did not choose to heal his hands. One is that he was interrupted and did not have enough of a chance to focus on healing his own hands. Another possibility is that, as Wong warned him that there would be serious consequences for using the eye, he chose not too. It's also plausable that he simply chose not to and had discovered his new calling. Doctor Strange was warned of the risks of using the time travel powers and also he had moved on and was not focused on helping himself by that point.
Doctor Strange did not use the time travel powers to heal his hands because Wong warned him of the dangers of doing this. He chose not to use the time powers to heal his hands because he had grown beyond self-interest and was immersed in his new calling as a sorcerer.