Projectile Atmospheric Entry Simulations
I put together a bit of code to compute the trajectory of projectiles as they fall through the Earth's atmosphere. I made the following simplifying assumptions:
All projectiles were fired from a circular orbit equal in altitude to the ISS (about $400~\text{km}$).
The atmosphere is non-rotating with no winds, and is identical to the standard atmosphere.
All projectiles fall perfectly straight: there are no lateral aerodynamic forces.
None of the projectiles' properties are altered by reentry.
I computed two scenarios for each projectile. The red trajectory is for a bullet fired towards nadir (straight down), and the blue trajectory is for a bullet fired retrograde (opposite the direction of orbit).
The 50-caliber Browning Machine Gun cartridge was originally developed to fulfill an anti-aircraft role, but later became a popular round for snipers. Its high mass helps it keep its speed and accuracy, even at distances of over one mile. In fact, more than half of the 15 longest sniper kills were made with the .50 BMG.
Projectile mass: $650~\text{gr}~(42~\text{g})$
Projectile diameter: $0.510~\text{in}~(13.0~\text{mm})$
Muzzle velocity: $3030~\text{fps}~(920~\text{m}/\text{s})$
Drag computed based on data (pdf) from the Ballistic Research Laboratory at Aberdeen (see figure 19).
(The dashed line shows the orbit that the projectiles were fired from.) The impact points are:
Retrograde: $2600~\text{mi}~(4180~\text{km})$ downrange in $12~\text{minutes,}~30~\text{seconds}$.
Nadir: $2050~\text{mi}~(3300~\text{km})$ downrange in $9~\text{minutes,}~15~\text{seconds}$.
(The dashed line is the lower boundary of the stratosphere at $12~\text{km}$, slightly above the cruising altitude of most airliners.) Note that both shots impact at a steep angle. Although the .50 is heavy compared to most reentry vehicles (in terms of ballistic coefficient) it still loses almost all of its velocity to drag and ends up pretty much falling at terminal velocity.
(Above the dashed line the speed is measured in kilometers per second; below the line, the speed is measured by Mach number.) Again, we see that most of the velocity is lost in the tenuous upper atmosphere.
This plot shows the projectiles' total energy loss rate. Not all of this energy will heat the projectile, however: in fact a good portion is used ionizing and heating the air the projectile encounters.
Calculating the peak heating according to this Institute for Defense Analysis document (pdf), the projectile will encounter peak temperatures of over $4000-5000~\text{K}$. Modern ablative heatshields for spacecraft are built to withstand up to around $2600~\text{K}$ (although classified heatshield technologies for ballistic missile warheads may have moderately better performance). This means that the projectile would surely disintegrate upon entry.
30×173 mm (Armor-piercing incendiary w/ DU penetrator, PGU-14/B)
This massive shell is used by several autocannons and chain guns, most notably the GAU-8/A Avenger: the primary armament of the Fairchild Republic A-10 Thunderbolt II "Warthog", an anti-tank air support and attack aircraft (and a personal favorite of mine). The projectile consists of an aluminum "jacket" surrounding a $10\frac{1}{2}~\text{oz}~(300~\text{g})$ depleted uranium penetrator.
Although typical accuracy of the GAU-8 autocannon is only 40 yards at 4000 yards distance, the projectile is probably capable of accuracy similar to typical sniping rounds if fired from an appropriate gun (although such a weapon may be too large for a single operator, and would certainly not be man-portable).
Projectile mass: $1~\text{lb}~8\frac{1}{2}~\text{oz}~(695~\text{g})$
Projectile diameter: $1.18~\text{in}~(30.0~\text{mm})$
Muzzle velocity: $3030~\text{fps}~(1010~\text{m}/\text{s})$
Drag computed based on data (pdf) presented by the University of Sarajevo (see figure 9).
The impact points are:
Retrograde: $2550~\text{mi}~(4100~\text{km})$ downrange in $10~\text{minutes,}~55~\text{seconds}$.
Nadir: $2030~\text{mi}~(3260~\text{km})$ downrange in $7~\text{minutes,}~50~\text{seconds}$.
Although the shots now have distinguishably different trajectories, again they both impact at nearly the same angle (although both are more horizontal than the .50).
This time the projectiles maintain their velocity down into the stratosphere, but are still limited to terminal velocity at impact.
With the bulk of the deceleration occurring in a denser portion of the atmosphere, peak temperatures are now on the order of $9\,000~\text{K}$.
The Armour-piercing fin-stabilized discarding-sabot round is a kinetic energy penetrator designed to defeat modern vehicle armor. For maximum penetration, the projectile is more dart-shaped than bullet-shaped. The projectile I found data on is the DM13, a non-DU round similar to the 120 mm M829 round fired by the United States' main battle tank, the M1 Abrams.
Since the fins stabilize the projectile in flight, APFSDS rounds are fired from smoothbore guns that allow increased muzzle velocity. This allows superior accuracy; modern tank crews can make a kill shot on another tank at several miles. But again it is unlikely that such a weapon could be operated by a single person.
Projectile mass: $9~\text{lb}~12~\text{oz}~(4423~\text{g})$
Projectile diameter: $1.50~\text{in}~(38~\text{mm})$
Muzzle velocity: $5000~\text{fps}~(1500~\text{m}/\text{s})$
Drag computed based on data (pdf) from the Ballistics Research Laboratory.
The impact points are:
Retrograde: $1950~\text{mi}~(3140~\text{km})$ downrange in $8~\text{minutes,}~40~\text{seconds}$.
Nadir: $1340~\text{mi}~(2150~\text{km})$ downrange in $4~\text{minutes,}~40~\text{seconds}$.
Both shots have extremely straight trajectories and impact at shallow angles, indicating that they have maintained speed down to the surface.
This time the projectiles impact at hypersonic speed, delivering a devastating blow beyond the capability of any modern non-explosive round.
However, peak temperatures are now on the order of $13\,000~\text{K}$. Even though heating only occurs for a few seconds (in-atmosphere the projectile loses around a kilometer of altitude every second) the intensity is so great that the projectile will be molten by the time it reaches the surface.
Postmortem
We can see that there is a fundamental tradeoff between heating and impact speed. The projectile must be large and lightweight in order to decelerate slowly and avoid burning up; however, it must also be heavy and dense to retain its speed. There is no middle ground between the two.
This means that a successful projectile would have to change aspect during its trajectory, and since we're limited to "a single chunk of metal, no stages or layers," this is not possible.
Another issue is accuracy. Since the projectile takes several minutes to hit the target, there is no possibility of a second, more accurate shot after using the first to 'scope in.' This, combined with the fact that the impact point is thousands of miles away means that all the targeting will be computerized.
The position of the shooter can be pretty precisely determined (within meters) by GPS (even on-orbit). GPS also gives precise timing. Star trackers can give milliarcsecond angular resolution with reaction wheels for pointing. The main challenge is aerodynamic perturbations:
The first problem is the orientation of the projectile. In the diagrams above, the projectile is shot right or down, but enters the atmosphere heading left. Thus the projectile would have to be shot backwards.
The second issue is stability. In the very thin upper atmosphere spin-stability will not work if the projectile is long and thin. Explorer-1 (the United States' first satellite) was designed to spin about its long axis, without accounting for the (mathematically difficult) dynamics of free rotation in 3D, and it predictably transitioned quickly into a 'flat spin.' This pretty much guarantees that an inert bullet would tumble as it enters the atmosphere, no matter how we try to stabilize it.
The third issue is wind: namely, the shooter would have to account for winds through the entire height of the atmosphere; and be able to predict the (highly chaotic) winds nearly ten minutes in advance.
An Alternative
If you're really set on shooting someone from space, you're going to need an active projectile. The projectile should consist of a $1.5~\text{m}$ long tungsten penetrator surrounded by a large sabot made of a lightweight insulating ceramic with an ablative coating. The sabot will need a large, flat front. Finally, the projectile should include a large solid motor and some variant of solid-state attitude thrusters.
First the projectile would be programmed with its target and released from the launch platform at a relatively low speed.
Once well-separated from the launcher, the solid motor will execute a deorbit burn.
As the projectile enters the atmosphere, the ablative coating protects the penetrator from heating.
Just after peak heating, the sabot (along with the solid motor) is jettisoned with pyrotechnic fasteners.
At this point, the penetrator begins to use aerodynamic maneuvering (like a missile) as it falls at its terminal velocity of Mach 2.
The penetrator uses GPS for terminal guidance and impacts within meters of the target.