[Tirned Tinnu]

Whoa! This thread took a bad turn. Let me see if I can help it.
As I see it, this is a read between the lines sort of scene.
It is more obvious to those of us that have read the Appendixes, and know that Aragorn had served under Ecthelion II, Denethor's father.

In Boromir's death, Aragorn has more to face than just Denethor's wrath. He has himself to pummel too! For if Boromir was anything like his grandfather, this would have been a terrible blow.
Aragorn's relationship with Ecthelion ended in Ecthelion dying of a wound from a morgul blade. Seeing Boromir die before him must have given Aragorn such a sense of deja vu that he must have suffered greatly!

Boromir's acceptance of Aragorn as his leader was written into the film, and was not actually in the books. He did ask Aragorn to take care of Minas Tirith, and the pledge that Aragorn gives him is a telling one. It meant more to Aragorn than Boromir could ever know, for Boromir did not know that Aragorn had been Throrogil, the captain of renown who battled with Ecthilion.

Don't you think it's interesting that Aragorn preferred to remain silent on this issue? If he had told Boromir who he had masqueraded as, Boromir surely would have been awed yet worried over Aragorn's part in this.
I found that the inclusion of Boromir's acceptance of Aragorn as King was a very sentimental touch added by people who did not fully understand Aragorn's relationship to Gondor. If they had added that Boromir knew of Aragorn's history, it would have made it much more powerful in the eyes of those who have read all of The Appendixes. I certainly would have been shocked out of my boots.
But that would be another story, and make the film just a bit too long. Alas for time allowances.