Deepak Chopra co-wrote The Seven Spiritual Laws of Superheroes with his son Gotham (an anglicization of the Buddha's original name - he wasn't named after the city Batman patrols). Chopra Sr. admits that he comes to the world of Western superheroes recently, by way of his son (a comics enthusiast and entrepreneur).
Chopra points out superheroes' similarity to mythological figures, manifestations of the collective unconscious. He gave his children Indian comics telling the tales of Krishna, Kali and other gods and goddesses. Similarly, I read the Bible as a comic growing up, years later connecting the fantastical feats and abilities of Moses, Samson, Elisha, Joshua and other figures with superheroes. This stuff is worth exploring. The Bible isn't common currency to many kids these days (outside of evangelical circles), nor are the myths of Greece and Rome, but the imagination responds to these modern incarnations of larger than life figures.
Unfortunately, Chopra doesn't have a good enough handle on the superheroes to really extrapolate their spiritual significance. Superman and Batman have been in constant publication since the 1930s, Spider-Man, Captain America, Thor, the X-Men, the Avengers and the Fantastic Four since the 60s, in issues that number into the hundreds, and in some cases, thousands. If you really want to get to the essence of a character, it's going to take some serious research.
The book's full of generalizations about superheroes as exemplifiers of various spiritual principles, admittedly adapted from Chopra's earlier work The Seven Spiritual Laws of Success:
Superheroes are not prisoners to the known.
Superheroes do not feel compassion when faced with adversity or someone who is suffering. They are compassion.
Superheroes understand that even violent behaviour is a cry for attention and therefore a cry for love.
Superheroes understand that creativity is not an impulse, but a process.
Superheroes see every challenge along the way to their intended outcome as an opportunity for creativity, but they never lose sight of the outcome.
The ultimate goal of superheroes is to reach unity consciousness, not intellectually, but experientially.
Statements like these can be found on most pages (they're excerpted above in isolation - they're all part of larger points in the book). Chopra seems to be using the term "superhero" as a substitute for "realized being." And these statements probably do apply readily to Superman. And Captain America. Maybe the Flash, too, and J'onn J'onzz, the Martian Manhunter. But that's about it. I can think of far more superheroes who are flawed as all people are, and often fall victim to the dictates of their personalities.
The spiritual significance of superheroes lies in their continued striving for an ideal, despite how often they fall short. They've dedicated their lives to a dangerous path, throwing themselves in the way of 
whatever danger presents itself as a sacrifice for people who don't reward them, don't know them, and in some cases, distrust, hate and persecute them. A book on this subject, grounded in multiple specific examples - Daredevil's nervous breakdown, Clint Barton's rebellious assassination attempt on Norman Osborn, Iron Man's alcoholism, and his Machievellian turn against many of his fellow heroes in Marvel's Civil War, Wolverine's violent past, Spiderwoman's alienation, Angel's suicide, Jessica Jones' ambivalence with authority and her occasional promiscuity, the Punisher's murderous justice, Green Arrow's paranoia and his human rights violating treatment of Dr. Light in Identity Crisis, the Hulk's possibly justifiable destruction of New York City in World War Hulk, and the activities of every member of the Watchmen - is one I look greatly look forward to reading. Or, if necessary, writing.
