We finally have our answer to the question whether Aragorn would appear in The Hobbit movies - No!
Viggo Mortensen recently told The Guardian that he turned down an offer to reprise his role in The Hobbit movies. Instead, he is looking into possible directing, moving on, as he is 54 years old. It would have been awesome if he returned to Middle-earth, but, oh well, he did not.
Correction: Viggo Mortensen was never asked specifically to reprise the role of Aragorn. Instead, he was asked if he was interested, to which he answered no. It is a small distinction, but I will admit it is an important one. I apologize for posting inaccurate information.
Source: The Gaurdian