Många har svarat på frågan om varför ingen var på ministeriet när Harry & hans vänner anländer - det var faktiskt timmar före 17:00 när de anlände, eftersom det var mycket planering, de gick igenom Förbjudna Skogen och hälften av London först, vakterna lurades eller hjärntvättas av eller arbetade för Voldemort etc. - men ingen har ännu svarat på den andra frågan OP-frågan, frågan om varför Voldemort inte kunde hämta profetian. Nu kanske han / hon borde ha skiljat dessa frågor, eftersom de inte är mycket relaterade, men oavsett faktum förblir det faktum att ingen har svarat på det.
When Trelawney first said the prophecy to Dumbledore, there was only one other person present, a Death Eater, Snape. However, Snape did not hear the entire prophecy. He only heard half, because he was caught spying halfway through. Voldemort did not care about the rest of the prophecy, as he thought he had heard all he needed.
Harry's repeated victories against Voldemort eventually forced him to reconsider. However, when the Department of Mysteries labeled the prophecy and stored it in a crystal ball somewhere, using sophisticated magic Rowling will probably never reveal even if she knows, the Department labeled the prophecy as belonging to Harry alone, rather than Voldemort or Neville (the prophecy could have been about Neville, but Voldemort "chose" Harry, which was an important part of the prophecy). I don't believe it was said whether prophecies could belong to more than 1 person, but perhaps the Department specifically chose only Harry as its pertainee because it did not want to give Voldemort any more ammunition. Therefore, for reasons I do not believe Rowling has explained, only Harry could pick up & view the prophecy, and others around him could see/hear it only if he did so.
In order for Voldemort to hear the rest of the prophecy, he had to either get Dumbledore to tell him (Trelawney does not remember her "true" prophecies, as illustrated in the same book), or he had to get Harry to hear the prophecy and tell him. He did not choose the Dumbledore route, for obvious reasons. There were 2 ways Harry could hear the prophecy: from Dumbledore, likely long out of earshot of Voldemort and without the desire to tell him later, or from the "Prophecy Ball" in the Department of Mysteries, within earshot of Voldemort if he planned it right. Again, he chose the Department of Mysteries, for obvious reasons.
Unfortunately for Voldemort, Harry was stronger-willed than he likely intended, and broke the ball rather than let him hear the prophecy, even though Harry knew he may never hear it himself. Voldemort was making it seem like it held mysteries as to why his parents were killed, among others, which he thought Harry couldn't resist (he couldn't), even though Voldemort likely knew they were utter nonsense (they were the reason Voldemort killed Harry's parents, but had no more importance than that), but Harry did resist them. This is probably part of the "courage" or "strength of character" Gryffindors are chosen and known for. Harry didn't know at the time that Dumbledore had heard the prophecy, which Voldemort was also counting on, but he had, so luckily Harry heard it later, but Voldemort never did. It probably wouldn't have done him much good, anyway, and knowing his obsession with patterns and complexity, it may have actually done him harm.