So it's been in the news recently that the US military has these rail guns that can hit a target a hundred miles away at approximately 4,500 MPH.  At 100 miles a way, the earth would have curved downward by 6100 feet.  I don't see how the relatively weak force of gravity (compared to a projectile being hurled at that velocity) could keep the projectile on target.

Why not?
Orbital velocity is the velocity at which the "falling" due to gravity equals the curvature of the orbit. At higher velocity, the object doesn't fall fast enough for a circular orbit.
The ISS velocity is about 17000 mph at 250 mi altitude. So the rail gun would be fine up to 17000 mph. (Actually a bit more since orbital velocity at sea level is a little higher.)
We can check this, by the way.
A projectile drops 32 ft/sec^2. Starting from zero vertical velocity, it would drop 6100 feet in about 20 seconds. Just due to gravity.
At 4500 mph, it takes 80 seconds to go 100 miles. If it were going 100 miles in 20 seconds rather than 80, then it would basically follow the earth's curve and be going orbital velocity. That would be 4x4500 mph or 18000 mph, which for the rounding I've done is close enough to the 17000 mph orbital velocity.

No, the projectiles are NOT in orbit.  Their velocity would clearly exceed the ability of gravity to pull it down to follow the curve over that distance.

Obviously they projectiles are not in orbit. That's the point. 
17000 mph is approximately orbital velocity at sea level. 

The projectiles going 4500 mph are going LESS than orbital velocity, therefore they DO fall back to earth.