The first assault rifles were developed during WW2 by Germany(StG44). Shortly after, the Soviet Union followed with the Kalashnikov guns and countries such as Belgium also started developing their guns. Why did the US not try to keep up with this arms race and only released it's first assault rifle in 1964(M16)? Also, what effect did this have on events such as the Korean War?
The M16 wasn't the first US assault rifle. The M2 carbine was adopted in 1944, and used in WW2. It's an assault rifle: intermediate cartridge, selective fire, 30 round detachable magazine.
If, by the Belgian gun, you mean the FN FAL (often not considered an assault rifle, since it uses a full-power 7.62mm rifle cartridge rather than an intermediate cartridge), then you also need to include the M14, which was adopted in 1957, following trials in the early 1950s, following development work on a replacement for the M1 Garand rifle which began in 1945.
Also, what effect did this have on events such as the Korean War?
The M2 and M3 carbines were the only assault rifles in significant use in the Korean War. The US was still using the M1 rifle (Garand) as its main infantry weapon, with the M2 filling the same role as submachine guns (SMGs) did in the other armies involved in the Korean War. If they hadn't been using the M2, they would have been using an SMG in the same role, with similar effect, so effect was limited.
The main reason why the US did not adopt a true Assult Rifle was the insistence on keeping .30-06. Staff in the US military became firm believers that any caliber used by the military must provide sufficient stopping power to kill an enemy soldiers and stop him dead quickly and powerfully. Thus there was pressure for any future rifle of the army to use the full 30-06 rifle cartridge. The ordnance department during the 20s and 30s was incredibly stubborn in this effort as well as they did not want to have to procure new ammunition when their .30-06 stocks were still good.
During the Rifle trials of the 30s there was an attempt to use .276. A lighter round that would have been easier to adapt into a semi and fully automatic rifle. Even though .276 passed trials the ordenence department blocked its future use in service rifles for fear have having to rid their .30-06 stock in favor of new ammunition. The underfunded army of the 30s could simply not afford to purchase all new rounds while .30 was still good.
The board of the army branchs in 1931 also refused the use of .276. stating "the importance of sunk costs and interchangeability with machine guns...". For the time being the US would be stuck with 30-06. Garand did end up developing a accurate semi-automatic rifle with the full power cartridge and its service in the second world war was successful enough for the US military to not consider its replacement until much longer after the war versus other nations whose primary rifles in the Second World War were bolt action rifles.
This is were my droning on about calibers becomes important. 30-06 had become so entrenched in the US that it was impossible in the 40s and 50s to replace. The M1 Garand performed fantastically in the war and merited no replacement. Other nations on the other hand especially in the Soviet Union had realized two things.
Both of these realizations had yet to thrust themselves on the US Military as the Semi-automatic rifle they had employed in the war had more suppression then the bolt actions that they fought against. And even though they new as early as 1919 that distances in combat where shrinking the US army still placed heavy emphasis on marksmanship over pure suppression.
Why Caliber is important is that .30-06 is too powerful to be put into a lightweight fully automatic weapon. Both the AK-74/AKM and StG-44 used smaller less powerful intermediate cartridges to make soldier controllable fire possible. In both Germany and The Soviet Union there was an acceptance that less power was needed for an Assault Rifle. The Germans where looking for a less powerful rifle that could put out more fire while the Soviets where looking for a more powerful and heavy hitting sub-machine gun. However US experiences with the Garand did not merit either of these thoughts and the semi-auto rifle proved just fine in the Second World War.
The US Refusal to leave .30-06 made this jump to Assault Rifles impossible. Forcing not only them but the rest of NATO to use slower firing 'Battle Rifles' could take the larger caliber and could theoretically fire in full auto but in almost all situations this was uncontrollable and the British went as far as to remove automatic fire from their rifles completely.
In summary all of the realizations (Combat distance, smaller caliber, suppression) required for the development of an assault rifle did not exist in the US military until the 60s where experienced in Viet-Nam had proved that large rifles such as the M14 with its only practice fire mode being semi automatic were not sufficient for modern combat.
Sources
Ezell, E. (1984). The Great Rifle Controversy.. Harrisburg: Stackpole Books.
Ezell, E. (1988). The AK47 story. Harrisburg, PA: Stackpole Books.
McNaugher, T. (1984). The M16 controversies. New York: Praeger.