I too was asking myself this question, and had to go back and watch the movie over again. Here's how I see the movie turning out without Indiana Jones' presence.
Regardless of whether Indiana led the Nazi's to Marion, they would have found her anyways. They already knew that Abner Ravenwood had the amulet, like the U.S. Army Intelligence officers told Indiana at the beginning of the movie. Marion is Abner's daughter, so it's only logical that they would have gone to see her eventually.
During the scene at Marion's bar, the Gestapo interrogator, Major Toht, would have tortured and most likely killed Marion, and would have obtained the amulet. Since the fight between Indiana and the bad guys would not have happened, her bar would not have caught fire. That combined with the Nazis already having the amulet, Major Toht wouldn't have gotten burned by the amulet (which is how the Nazis only got one side of the amulet), and thus would have looked in the right place. The Nazis would have gone on to open the Ark just as they did and every one would have died, just like Amy says. Finally, the Ark would have been left at the site where they opened it for someone else to discover, so who knows what would have happened after that. You could say Hitler would have sent more Nazis to investigate, or maybe the U.S. would have found it and suffered the same fate.
The one major thing here is that if you at any point remove Indiana from the story, the Nazis would have killed and/or tortured Marion. So, for that, Indiana Jones is still a hero!
As for how Indiana Jones got the Ark off of the island, the answer isn't really revealed in the movie, but the most logical conclusion I have is that he stole the submarine that the Nazis used to travel to the island.
Note: I don't remember where I saw it (might have been in TBBT), but there was a comment that without Indiana Jones, the Ark would have been opened in front of Hitler. That's not true, because when they get to the island, Belloq gives a speach about making sure that it was "the one true Ark" before handing it over to Hitler (for fear of what Hitler would do to them if it wasn't). So, he would have opened it before it got to Hitler anyways.