A number of England fans stormed Wembley Stadium during the Euro 2020 final against Italy, causing the Football Association (FA) to be handed a two-match ban, one of which is suspended.
England face Italy again on Saturday at Molineux, with only 3,000 fans allowed to attend, most of which will be children.
"If it is an embarrassment, it is for England as a country," Southgate told a news conference on Friday.
"A lot of the people that caused the problems I'm not certain were football fans.
"We spoke enough about it, we spoke about it after the final and when the punishment was first given, what I will say is the vast majority of our fans who travelled to Germany on Tuesday behaved brilliantly, a big thank you to them because maybe people were thinking something different but there were a huge majority who were a credit."
England are at the bottom of League A Group 3 in the Nations League with one point after losing 1-0 to Hungary and drawing 1-1 with Germany. Italy, who have failed to qualify for this year's World Cup in Qatar, are top with four points.
"We are still assessing both," Southgate said.
"Phil joined us in the training this morning so he's progressing. Not quickly, so there's some doubt as to whether he can be involved in either game. He's obviously keen so we will try to give him as long as we can.
"Kalvin didn't train with the group so he would be doubtful for tomorrow but we will see in the morning."
"I think this is a good game for some of the other players to come into it. It's a top-level game and a good sign to trust the players.
"If we give players only certain types of matches, then you don't find enough about them and they don't find the tactical challenges.
"For us, it's a balance always knowing that we've got to win matches but also try to find out about the players and them getting good experiences."
Information from Reuters contributed to this report.