You've reminded me of one of the major changes in the movie, from the book. In the movie, Aragorn finds Boromir before he dies, hears his confession, and forgives him for trying to take the ring. In the book, Boromir dies alone, failing to rescue Merry and Pippin. He never has a chance to confess.
I'm not Catholic, but I know JRRT was, and I'm sure the fact that he never gave Boromir a chance to "confess" may have been significant. I'm not sure which works better though, the movie or the book.
A mile, maybe, from Parth Galen in a little glade not far from the lake he found Boromir. He was sitting with his back to a great tree, as if he was resting. But Aragorn saw that he was pierced with many black-feathered arrows; his sword was still in his hand, but it was broken near the hilt; his horn cloven in two was at his side. Many Orcs lay slain, piled all about him and at his feet.Aragorn knelt beside him. Boromir opened his eyes and strove to speak. At last slow words came. 'I tried to take the Ring from Frodo ' he said. 'I am sorry. I have paid.' His glance strayed to his fallen enemies; twenty at least lay there. 'They have gone: the Halflings: the Orcs have taken them. I think they are not dead. Orcs bound them.' He paused and his eyes closed wearily. After a moment he spoke again.
'Farewell, Aragorn! Go to Minas Tirith and save my people! I have failed.'
'No!' said Aragorn, taking his hand and kissing his brow. 'You have conquered. Few have gained such a victory. Be at peace! Minas Tirith shall not fall!'
Boromir smiled.
'Which way did they go? Was Frodo there?' said Aragorn.
But Boromir did not speak again.
'Alas!' said Aragorn. 'Thus passes the heir of Denethor, Lord of the Tower of Guard! This is a bitter end. Now the Company is all in ruin. It is I that have failed. Vain was Gandalf's trust in me. What shall I do now? Boromir has laid it on me to go to Minas Tirith, and my heart desires it; but where are the Ring and the Bearer? How shall I find them and save the Quest from disaster?'
He knelt for a while, bent with weeping, still clasping Boromir's hand. So it was that Legolas and Gimli found him. They came from the western slopes of the hill, silently, creeping through the trees as if they were hunting. Gimli had his axe in hand, and Legolas his long knife: all his arrows were spent. When they came into the glade they halted in amazement; and then they stood a moment with heads bowed in grief, for it seemed to them plain what had happened.
One thing that occurs to me in the time since Gandalf's passing is that Tolkien's book Aragorn is a much more self-doubting leader than Aragorn has expressed so far in the film.
Granted, much of the self-doubting of the book is internal thought, difficult to show on film. I am interested to see if PJ brings any of Aragorn's insecurities out in TTT... or if he allows him to remain heroic and confident on the outside, without all the doubts we know he has from the book.