Han Solo didn't really deserve a medal after the Battle of Yavin in the first Star Wars film. The Battle of Yavin was the first major battle of the Galactic Civil War, which saw the Rebel Alliance lead a squadron of X-Wing fighter pilots to destroy the original Death Star. Luke Skywalker was the star of the show, having been abandoned by Han who was more interested in his payout from previously helping him and Ben Kenobi. Late in the battle, though, Solo's conscience pangs led him to swoop in and save the day, and it proved that the smuggler finally did the right thing.
That said, his contribution to the whole fight was quite minor. Han returned to the Battle of Yavin with great reluctance, only after many Rebel pilots had already lost their lives and most of the enemy threat was eliminated. He also didn't face the Death Star's long, narrow trench run, where the Imperials provided constant pressure to their opponents with repeated cannon fire. Solo simply arrived at the last minute, only to destroy a single TIE fighter, and flew off with the gang before the enemy base exploded. As a result, it's extremely unfair that he and Luke are on equal standing during the ceremony as Luke did most of the work and demanded nothing in return.
Han Actually Refused His Medal In A New Hope
Star Wars: From the Adventures of Luke Skywalker, a Legends novel by Alan Dean Foster showed a different version of the Rebel Alliance ceremony seen in A New Hope. In the book, Han Solo was humbler than usual, initially very reluctant to accept the Medal of Bravery. Star Wars' Han Solo felt that the fighter pilots who lost their lives in battle and never made it back should have been awarded the great military honor instead.
Eventually, though, after being convinced by Princess Leia, Han took part in the grand ceremony to be celebrated for his part in the battle. By having Solo be more reluctant with accepting the gold medal, audiences would have briefly gotten to see a new side to his character never before shown on-screen. This version of the ceremony would also prove to be much more satisfying, given that his contribution to the battle was very minor.
Why Chewbacca Didn't Receive A Medal After The Battle Of Yavin
Even to this day, Star Wars fans have always made the argument: while Chewbacca's contribution to the Battle of Yavin was minor, if Han deserved a medal, he should also have received one. However, in the years since A New Hope's theatrical release, director George Lucas explained why Chewbacca didn't get a medal during the grand Rebel Alliance ceremony, claiming that "they don't really mean much to Wookiees". Instead, the furry creatures have their own celebrations, and "Chewbacca was in fact given a great prize and honor during a ceremony with his own people."
While audiences never saw the Wookiee get a medal after the Battle of Yavin, an alternate version shows that he did in fact receive one off-screen in the 1980 Star Wars comic, "The Day After The Death Star". Here, a panel of the book featured Princess Leia standing on a table to bestow a medal upon the Wookiee a day after the Death Star battle. Although Chewie's recognition of heroism wasn't shown on-screen, it's great to know that he was in fact rewarded for his bravery. That said, a positive to Star Wars canon is that it offers many alternate versions of the live-action films, which enables fans to envision their own way of how things should or shouldn't have gone.