Mboyo, a 25-year-old striker who has played twice for Belgium, had the chance to level the score in the 15th minute of Saturday's match at home to Waasland-Beveren, but gently side-footed the spot kick wide.
Mboyo confronted fans at the end of the match, which Ghent lost 2-0, and appeared to lose his cool.
During an unscheduled news conference on Monday, Mboyo said he was sorry and asked for Ghent fans to accept his apology, although he said his post-match outburst had been provoked by racist comments from a few supporters.
"Maybe I would not have reacted if I wasn't captain. To show that I shouldn't have done so, I will no long wear the captain's armband," he said.
Ghent are without a win in eight matches. They have a Belgian Cup quarter-final tie on Tuesday against league leaders Anderlecht, who thrashed them 5-0 in October.