The video of the execution in the eastern city of Derna was posted on social media websites, including YouTube.
Amnesty International said it was carried out on August 19, apparently by an extremist militant group,the Shura Council of Islamic Youth, in front of a crowd seated in the stadium.
"This unlawful killing realises the greatest fears of ordinary Libyans, who in parts of the country find themselves caught between ruthless armed groups and a failed state," said Hassiba Hadj Sahraoui, Amnesty International's Middle East and North Africa Deputy Director.
"Such acts can only lead to further human rights abuses in Derna, where residents have no recourse to state institutions and therefore no means to seek justice or effective protection from abuses," she added.
"The Libyan authorities, with the support of the international community, must urgently address the breakdown of law and order that has persisted in Derna and elsewhere following the end of Colonel Kadhafi's rule."
In the video, the Egyptian, named as Mohamed Ahmed Mohamed, is brought into the stadium blindfolded on a truck.
A gun is then passed to an unmasked man wearing plainclothes, believed to be the brother of the dead Libyan, who then shoots the Egyptian from behind.
Since the fall of long-time dictator Moamer Kadhafi in a NATO-backed 2011 uprising, interim authorities have failed to establish order and security in a country prone to anarchy and deadly violence.
They have been unable to restrain a large number of militias formed by ex-rebels who fought Kadhafi and who still hold sway across Libya.
Since mid-July, Libya has been rocked by fierce and deadly fighting between militias that has prompted an exodus of foreign nationals from the oil-rich North African country.
Amnesty said armed Islamists in Derna "appear to have taken advantage of the breakdown of the rule of law to assert their control in an apparent attempt to enforce their own interpretation" of Islamic sharia laws.
Get Breaking news, live coverage, and Latest News from India and around the world on NDTV.com. Catch all the Live TV action on NDTV 24x7 and NDTV India. Like us on Facebook or follow us on Twitter and Instagram for latest news and live news updates.