Clearly not. If a printer runs out of ink, does that mean it is no longer a printer? If a car runs out of gas, is it no longer a car?
If I am printing copies of the Koran, and the government decides to tax ink at the rate of 17,000%, are they just imposing levies, or are they restricting my first amendment? What if the ink tax applies to everybody? What if the ink tax only applies to publishers printing things that the government doesn't like?
It's worth noting that while I've reference DC v Heller a number of times in this topic, it applies here as well. The DC handgun ban used to be a de facto ban on all handguns, but above and beyond that, even for rifles or shotguns that may have been legal to possess, there was a mandate that in the home they could not be used for self defense, and they must have been disassembled as much as possible, and ammo stored separately.
The Supreme Court struck down this requirement, with language in the sentencing stating that any law which would make a weapon impossible to use for self defense would also be unconstitutional. As such, allowing people to possess firearms, but not ammunition would be a violation of one's Constitutional rights to bear arms.
Also, to answer the question more directly, a firearm need not even be capable of firing to be considered a firearm in US law. There are parts one can buy that are 'controlled', and that means that those parts effectively comprise the gun itself. In the case of a Colt 1911 pistol, the 'frame' is considered the gun, even though it does not possess a trigger, or grips, or a barrel, etc. In the case of an AR15, a stripped lower receiver is considered the firearm, despite that it has no moving parts.
Knowing this, there are ways to circumvent FFL required registration as well. I can buy what's called an '80%' lower receiver, which has been machined, basically, to an outline, but has not had the internal machining done. The ATF considers this a paperweight, effectively, and I can buy an unlimited supply of these without notifying anyone. To complete them requires a drill press and some know-how, and legally, I may build these (without anyone else's assistance) and lawfully possess them as completed firearms without registering them federally (local statutes vary) and without requiring them to have serial numbers. I may not however, manufacture them with the intent to sell, and there are limits on how many of these I may build in a year (I think it's 50, but it's been awhile since I last looked).