Provided there was enough of the bullet remaining to analyze, there are a few things they could determine from it. They can measure the caliber and weight, and examine the bullet design to see who made it. So they could tell if it was a Federal HydraShok, a Remington Golden Saber, or perhaps what brand of lead bullet most closely matched the lead alloy used.
Next, they can measure the number of grooves and their angle if there is enough of the bullet left to do this. While it will not identify a specific gun at this time, it will eliminate certain makes and models. They may be able to say that the gun was a Smith and Wesson K frame, as an example. There may be several models possible, since more than one model can have the same number of grooves and the same twist rate.
Once they have an idea what kind of gun was fired, they could ask the gun stores for permission to examine their records to see who bought one of these. If the FFL agrees to let them do it, they need no warrant.
If they find that a similar gun was sold to a person of interest in the case, they can go and ask this person to examine his gun. They can get a warrant to seize it and test it if they have sufficient evidence.
And somewhere in THR's archives is the story of a man whose pistol fired a shell casing found at a drive by shooting in New York State. The police matched the casing and the gun, then learned that the owner's permit was revoked, but they never collected his guns. So they went to his house and asked him where he kept the gun. He told them where he had stored and they went to look for it.
But it was gone!
When they returned to the man's house to confront him with this new fact, he told the police that he wanted to report a stolen handgun.