@GrinningX is on the right track. I'll add detail based on existing bulletproof materials and standards.
What you're looking for is NIJ Level IV protection. Level III will stop a 7.62mm rifle round such as fired from an AK-47. With about 2000J of energy it's the most powerful bullet Cap is likely to encounter on a modern battlefield.
But Captain America's shield was for WWII. Bullets were a lot more powerful back then. He'd be facing things like .30-06 Springfield used by the US in WWII or 7.92 x 57mm Mauser used by the Germans. These are in the 4000J energy range. For that you need Level IV protection.
Then there's the V0 rating, the most energy a bullet can have and never penetrate. For example, if we say Cap's shield has a V0 for .30-06 then it means you can fire .30-06 at it all day and none will penetrate. This is a tall order. Each high energy impact will make small bends and cracks in armor and they will add up.
The more realistic measure is the V50 rating, the most energy a bullet can have an only penetrate 50% of the time. 50% doesn't sound like great odds when it's your chest on the line, and nothing about Captain America is 50%.
Most body armor is made up of layers to absorb and distribute the shock of impact, a combination of soft armor like Kevlar and hard armor like steel and titanium, all backed up by your squishy flesh. Cap's shield has none of these advantages. It acts as a "stand alone plate" meaning it will protect with nothing behind it.
- A stand alone plate
- Roughly 75 cm in diameter
- Providing Level IV protection
- With a V0 at .30-06
- And throwable by an Olympian
This is a tall order.
The lightest stand alone level IV plates I've looked at weigh about 4 to 5g per cm2. That means a 4600 cm2 shield will weigh 18 to 23 kg, significantly more than the comic's 5.5 kg. Worse, these are ceramic and only guaranteed to stop ONE level IV shot. Captain America cannot have his shield shatter after one shot.
This isn't going to work.
In order to make this work we're going to have to lower our requirement to level III and ease up on the V0 requirement. Sorry Cap, you'll have to hope the Germans have submachine guns and StG 44 assault rifles and do some ducking. Or maybe this is a modern Captain America.
You can get a titanium-steel multi-hit hard armor level III stand alone rifle plate. They're about 1cm thick and still 4 to 5g/cm2. This still leaves Cap with a 20kg shield.
For comparison, regulation Men's Discus uses a 2kg disc. This shield is ten times that.
Regulation Men's Shot Put is 7kg, still three times less than our shield.
To get into the right scale we need to look at the Highland Games. The heavy weight ball (on a chain) is 4 stone (25.4kg). Finally, we're in the ballpark. The Irish record is 9 meters. An impressive distance in a sporting event, but not a terribly effective ranged weapon.
Unless he hopes the bad guys will laugh themselves stupid as he heaves his only protection 10 yards towards them, I think Cap is better off using the shield as a shield and shooting the bad guys with a pistol.
A M240 machine gun weighs about 15kg loaded. It's considered very heavy (the M249 weighs 10kg). If you want to lug one around you generally do it with a sling.
It's still lighter than our shield.
Cap is expected to hold something even heavier with one arm, run and jump with it, move it quickly to block shots, and swing it to clobber enemies.
Make The Shield Smaller
We're not going to get any lighter material. At this point you'll have to cut down on the size of the shield and probably make it square. A more reasonable a 50 cm x 50 cm gives a 12.5 kg shield still wide enough to fully cover Cap's broad shoulders.
This is still very heavy and unwieldy, but puts it in the range of a very heavy automatic weapon.
There's good news for the throw. This puts it closer to the 35 lb (15.9 kg) weight throw, the record for which is 25 meters. Still, that's probably at a high arc and not a graceful Frisbee, unlikely to be terribly useful in combat. It's not going to bounce back, so he's thrown away his protection.
Still probably better off with a pistol.