Samir Sumaidaie said his 21-year-old cousin was shot as he helped marines who were carrying out searches at his village in the restive Anbar province.
Mr Sumaidaie said the ramifications of such a "serious crime" were enormous for both the US and Iraq.
US officials said the allegations would be thoroughly investigated.
Meanwhile, a suicide bomber has killed at least 20 people outside a special police recruiting centre in the capital Baghdad.
It is the latest in a spate of attacks targeting the country's security forces.
English exercise
In a letter to colleagues, Mr Sumaidaie explained in detail what happened to his cousin Mohammed al-Sumaidaie on 25 June in the village of al-Sheikh Hadid.
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All indications point to a killing of an unarmed innocent civilian - a cold blooded murder
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He said Mohammed, an engineering student, was visiting his family home when some 10 marines with an Egyptian interpreter knocked on the door at 1000 local time.
He opened the door to them and was "happy to exercise some of his English", said the ambassador.
When asked if there were any weapons in the house, Mohammed took the marines to a room where there was a rifle with no live ammunition.
It was the last the family saw him alive. Shortly after, another brother was dragged out and beaten and the family was ordered to wait outside.
As the marines left "smiling at each other" an hour later, the interpreter told the mother they had killed Mohammed, said Mr Sumaidaie.
"In the bedroom, Mohammed was found dead and laying in a clotted pool of his blood. A single bullet had penetrated his neck."
The US military said the allegations "roughly correspond to an incident involving coalition forces on that day and in that general location".
Maj Gen Stephen T Johnson said the allegations were being taken seriously and would be thoroughly investigated.
Acting US ambassador to the UN, Anne Patterson, had "expressed her heartfelt condolences" to Mr Sumaidaie, said a spokesman.
She has urged the Pentagon and state department to look into the matter immediately.
"All indications point to a killing of an unarmed innocent civilian - a cold blooded murder," said Mr Sumaidaie in his letter.
"I believe this killing must be investigated in a credible and convincingly fair way to ensure that justice is done, and the sense of grievance is mitigated, and to deter similar actions in the future."