It came after the Liverpool defender was involved in a bust-up with Sterling in the canteen at St. George's Park on Monday - an incident that resulted in Southgate axing the Manchester City forward from the squad for Thursday night's win.
"Joe hasn't done anything wrong & for me to see someone who keeps his head down and work hard especially after a difficult week for him to be booed when he came on tonight was wrong".
"I've taken full responsibility and accepted the effect".
United's most capped England player while with the club, Sir Bobby surely has a claim to be one of the greatest-ever to pull on the white shirt.
Asked whether he will speak to Gomez about the issue on Friday, Southgate said: "Absolutely". We're a team and Joe has done nothing wrong.
"I'm hugely disappointed for him, but he's got the support of all of us - the whole dressing room, totally united".
It wouldn't be accurate to say France were at their blistering best for the remaining 81 minutes, but they grew into the game with Raphael Varane's header coming at the ideal moment to level the scores after 35 minutes of play. "The gaffer spoke to me and a few of the other senior players in the team, and that's the decision we came to".
After the match Harry Kane revealed the players did have a say on whether to axe Sterling from the squad for the match, despite Southgate insisting it was exclusively his decision earlier this week.
Kane welcomed Sterling's imminent return to the team. "We move on. He has been good around camp and training well".
"Of course he would have been disappointed not too play today, but I'm sure he'll be ready to go and try and get on a pitch on Sunday".
A Harry Kane hat-trick and strikes from Alex Oxlade-Chamberlain, Marcus Rashford and Tammy Abraham, plus an Aleksandar Sofranac own goal, saw England qualify for next summer's tournament in style.