The nomination, published on a US government website, comes amid strained ties between the two nations.
Trump has been critical of South Africa on several fronts, including for a case it brought before the International Court of Justice accusing Israel, a top American ally, of committing genocide against Palestinians in the Gaza Strip.
In an executive order, he halted US foreign assistance to South Africa and declared that the US refugee system would give priority to Afrikaner "victims of unjust racial discrimination", claiming that South Africa was expropriating their land.
The South African authorities have not confiscated any private land since the end of apartheid in 1994.
Bozell is the founder of the Media Research Center, which says it works to "expose and counter the leftist bias of the national news media", according to its website.