This would be a difficult position to uphold. The elements of Shakespeare's tragedies, with some few variations, are derived from and illustrate the principles of tragedy as set out by Aristotle.
One principle is that the hero's inner trait flaw--or--flawed decision leads to his tragic end. You can see this principle at work in Hamlet, Macbeth and King Lear. Each hero makes a flawed decision (often based upon a inner trait flaw) that leads to their tragedy. For example, Lear makes the decision to foolishly rely on public demonstrations of affection from his daughters to determine their inheritance (he probably meant well but did not think well).
Another principle is that of the paramount importance of plot. Aristotle asserted and Shakespeare upheld the principle that plot--action--was the driving vehicle of tragedy. Plot actions must be connected in cause-and-effect relationship: Hero acts such a way that causes such an event, which causes such a responding action, which causes such a consequent event.
Tragedy is the imitation of an action; and an action implies personal agents, who necessarily possess certain distinctive qualities both of character and thought; for it is by these that we qualify actions themselves, and these--thought and [inner] character--are the two natural causes from which actions spring, and on actions again all success or failure depends. (Aristotle, Poetics Part I)
A third principle is less a principle than a consistent element of Shakespearean tragedy. A tragedy incorporates how the hero's inner flaw or flawed decision affects other characters. For Lear, his flawed decisions affected Cordellia's death. For Hamlet, his flawed decision (or flawed character, depending upon which critic you agree with) affected the deaths of his enemies and his mother and his own death. For Macbeth, his flawed decisions affected the murder of Duncan and myriad others.
Applying these concepts of Shakespearean tragedy to Chris McCandless in Into the Wild has its difficulties. For one thing, the death he suffered did not necessarily spring from flawed inner traits or flawed decisions. Penn and author Jon Krakauer make cases for Chris's death being accidental. One might say that his flawed decision was to tackle a winter in Alaska without better preparation. One might say that his inner trait flaw was a jaundiced and scarred reaction to materialism and violent relationships. Yet, in the final analysis, his death appears to have been an accident. This does not represent the depth of the flaws demonstrated by Shakespeare's tragic heroes.
Further, Chris's story is the result of spontaneous wanderings. As such, it is hard to assert that there is a developed plot based upon cause-and-effect actions, results and reactions. Chris's death in Alaska seems by all analyses of events to have been driven by desire for immersion into nature rather than the final link in an intentional and decision propelled cause-and-effect chain of action and effect.
Finally, while Chris was personable and sociable, he was at the same time reclusive with strong tendencies to condemn and reject society. He certainly made efforts to distance himself from his family and from friends as he made them in his wanderings. Consequently, it is hard to assert that his tragedy had tragic effects of a Shakespearean magnitude for other characters, except for natural parental grief and the natural grief of friends. In the end analysis, it is very difficult to find any substantial case for asserting that Into the Wild can be considered a Shakespearean tragedy.