Belize says its troops acted in self-defence when the Guatemalan tried to attack an army patrol with a machete, but Guatemala contests that.
The two Central American neighbours have a long-standing border dispute.
Last month, Guatemala deployed 3,000 troops along the frontier following a shooting incident in which a Guatemalan teenager was killed.
Belize said the presence of large numbers of Guatemalan troops along the border had increased tensions between the two countries.
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In a statement, the government of Belize said one of the three detainees had acted "aggressively" and ignored a warning shot before being shot and wounded in the elbow.
Guatemala`s foreign ministry rejected Belize`s version of events.
It said that one person with a machete could not attack a "properly equipped and armed" patrol.
The territorial dispute between the two nations dates back more than 150 years.
Belize became independent from Britain in 1981. However Guatemala did not recognise it for another decade because of the land dispute.
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