He is immortal, kept alive by the power of the Holy Grail. Presumably you don't have to eat or drink, in Indiana Jones, why was Henry Jones Sr. the Knight of Indiana Jones and the Last Crusade? When Indy's father appears in flashbacks at first. No, he said the Grail can't come out.
But it does imply that, in order to achieve immortality, one would have to drink continuously from it. So yes, Indiana Jones's father died and no, he wasn't immortal. The location of the grail is within a field where, as long as you remain in it after drinking the grail, you are immortal. So the moment you cross the seal, immortality is emptied.
For years, I believed Indiana Jones was immortal. I saw Indiana Jones and the Last Crusade when it first came out in theaters and, since I had a drink of the Holy Grail before leaving for the sunset, I thought Indiana Jones would live forever. It wasn't until years later that I realized that the grail's explanation of how eternal life was limited to the temple. And I'm telling you all this because it's just one example of why I'm still completely fascinated by the ending of Indiana Jones and the Last Crusade.
A theory explained (via CBR) believes that Indiana Jones' encounter with the Holy Grail may have given him supernatural powers. Indiana Jones (Harrison Ford) and his father, Henry (Sean Connery), have been searching for the Holy Grail throughout the film. While there is evidence that the Grail may not have made Indiana Jones truly immortal, it does appear to have given him some kind of restorative power. Indiana recovered the Grail to heal a gunshot wound that Donovan had inflicted on Henry Jones Sr.
(being hit by a car or dying in a plane crash), but other than those situations, Henry Jones, Sr., should not. They lived in the sanctuary of the cup, where a Great Seal acted to prevent the artifact from being carried beyond the entrance of the temple for nearly one hundred and fifty years. Schneider, who had almost fallen into herself, was saved by Indiana, but eagerly tried to grab the Grail below with her arm free. In 1938, Donovan contacted Indiana to pick up where Henry left off and find the Grail, which Jones used as an opportunity to search for his father.
Indiana, like Schneider, tried to reach the cup, but Henry persuaded her to let it go, thus avoiding Schneider's fate. In late 1937, Donovan hired Henry Jones to search for the Grail with Dr. Elsa Schneider, but both Donovan and Schneider were secretly allied with the Nazis, who had their sights set on acquiring the Holy Grail. The fourth film introduced elements of science fiction by having Jones deliver an alien's crystal skull to his ship in the ruins of an ancient city.
The first film, In Search of the Lost Ark, saw Indiana Jones recover and witness first-hand the power of the Ark of the Covenant. After being attacked by the Brotherhood of the Cruciform Sword seeking to protect the Grail, its leader, Kazim, told Indiana where Henry Jones was being held. I thought about it a lot and if anything could have gotten me out of my retirement it would have been an Indiana Jones movie.