Imad Kamal Sadeq, 49, and his 21-year-old son, David, were killed late on Wednesday after a row on a construction site near a church that the policeman was guarding.
Reuters are reporting that the policeman was in custody after authorities reviewed CCTV footage from the church's security cameras.
The construction site is in Egypt's Minya province, 75 miles south of Cairo.
Coptic Christians are calling for the state to provide more protection to believers in the area.
They say that they suffer discrimination from Egypt's Islamic majority which fuels attacks by militants.
The head of the local diocese, Archbishop Macarius, who led the funeral prayers, demanded a response from authorities.
He said: "We call for all armed police officers assigned to guard churches to be checked. Are they qualified to carry live ammunition, so that they don't become a source of danger rather than protection."
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