England were ordered to play their next two home games in UEFA competition without supporters present after disturbances during their Euro 2020 final last year.
Southgate's side host Italy in the Nations League on June 11, in a rematch of that final, behind closed doors. The ban on fans attending the second game has been suspended for a probationary period of two years.
"We're on a yellow card and we've got the embarrassment now of playing behind closed doors at home," Southgate said.
