Been asked before here - your question specially is the second question comment down. It’s all a good read.

https://www.reddit.com/r/AskHistorians/comments/2iy89g/why_was_the_556mm_round_chosen_as_the_nato/

I can't comment on the veracity of the claim, but I can provide some knowledge of international law. The Additional Protocol I to the Geneva Conventions, which has not been ratified by the United States, says in Article 35(2):

"It is prohibited to employ weapons, projectiles and material and methods of warfare of a nature to cause superfluous injury or unnecessary suffering."

This would seem to include maiming rather than killing soldiers, and may form part of customary international law. However, there are other questions explored in the body of international law. An ICRC report from 1973 (before the Additional Protocol I) actually tackles the question of small-caliber projectiles. The report notes that the effects of weapons firing small-caliber projectiles, particularly at high velocities, may resemble "dum-dum" bullets, otherwise known as expanding bullets, which are expressly outlawed in international law under the 1899 Hague Convention's Declaration 3 on the subject. This has led to numerous criticisms, including during the Vietnam War (a complaint was lodged by Sweden) and later, that the 5.56 bullet is illegal under international law. Sweden attempted to call for a small-caliber protocol as well, but the UN-sponsored Diplomatic Conference that resulted went nowhere on the subject, and no such protocol was passed. If the bullet is indeed illegal, it has not been judged as such yet by any relevant authority of international law, but there is reason to believe that it is illegal due to the bounds of military necessity not necessarily applying to the reasons for choosing it (i.e. ability to carry more rounds). The question is muddied however by the lack of clarity in research by other organizations and states on whether or not a better judgment could be made based not on caliber but on velocity in combination with that caliber, which complicates the issue somewhat.

The decision to use an intermediate round like 5.56 came from lessons learned during WWII. Every military on the planet began the war outfitting their troops with full-size rifles with effective ranges of 500+ yards firing .30 calibre ammunition. By the 1950s many countries (the USSR being a notable exception) were developing semi- and full-automatic full-size rifles with the aim of increasing soldier's fire output. However they were still limited by the full-size cartridges they fired, being heavy in both weight and recoil.

Analysis of western front firefights by Project SALVO concluded that the majority of combat took place after chance encounters with the enemy at relatively short range (under 300 yards) and that the primary determinant of the winner of a firefight was the number of rounds fired. The conclusion of the Project was that the most desirable infantry weapon fired a smaller-calibre cartridge accurately at a high rate of fire.

Partly as a result of this, the US Army began making requests for manufacturers to develop a rifle that had these qualities, eventually resulting in the AR-15 and the progenitor of the 5.56 NATO cartridge, .223 Remington, in the late fifties. It was not immediately adopted, however, as the US Army had just adopted the M14 and some members of the US military were skeptical that a smaller cartridge would be sufficient.

Then Vietnam happened, and more of the same lessons came back, and the AR-15 was finally officially adopted by the US Army as the M16, being used in a commercial off-the-shelf capacity by MACV SOG for years previous.

Basically, the US Army figured out after three wars that it's better to shoot more than to shoot big. 5.56 NATO is less than half the weight of 7.62 NATO and consequently soldiers can carry twice as much.