Would Jaime Lannister Apologize to Bran Stark if He Could?
Would Jaime Lannister Apologize to Bran Stark if He Could?
No. It doesn't mean that Jaime doesn't regret what he did. But he is not going to apologize. In fact, it would be unwise to do so.
Jaime now understands what it is like to be crippled.
According to A Storm of Swords, Tywin Lannister vows revenge for Jaime's hand even when Jaime tells him, I can hardly dress myself in the morning.
The eventual death of Vargo Hoat does not change Jaime's circumstances. The injury remains. This also goes for the show culprit Locke, who dies at the Wall. Jaime as yet does not know, but what good would an apology do for him? Would he even believe an apology from Hoat or Locke? Of course not.
Now, Let's Look at Bran
Right now, he does not remember that Jaime pushed him. We will eventually know if he is meant to remember the source of his affliction. By the time that happens, Bran may have already forgiven Jaime. But there are reasons he isn't remembering: trauma is one, and possibly magic is another. Knowledge of Jaime's deed may lessen Bran's abilities as a greenseer. I doubt Bloodraven plans to let that happen.
If anything, an apology will feel false and stilted. Jaime now understands through experience what he did. Just as he cannot get his hand back, he cannot heal Bran. Jaime may have too much respect for Bran's suffering to demean him with an apology. Instead, Jaime may show that he regrets his past deeds through what he does in the present. He is trying to help Sansa Stark by sending out Brienne in the books and show. In the books, he wants to be just in his dealings with Riverrun. He wants his honor back. It takes more than words to do that. It takes deeds. Jaime is all about getting things done.
So, it is highly possible that some accord may be reached between Bran and Jaime, as they became linked the day Jaime pushed him. GRRM may want to complete that thread, in fact, I am counting on it. But Jaime understands now what he did. He cannot apologize for taking away one's ability to walk. He can only atone for it and show that he is no longer that man who felt the need to push a small boy because he watched him having sex with his sister. He has to become a better man. Book Jaime seems to be doing just that. I think the show wants the same, but it is taking the long route.