Both the Football Association and Saka's club Arsenal issued statements on Monday morning condemning the abuse before Southgate addressed reporters in a post-tournament debrief via Zoom after England's agonising defeat in their first tournament showpiece for 55 years.
"I'm not totally across everything but my first thoughts this morning are immediately with the boys that have done so well for us," Southgate said.
"The players have had an incredible togetherness and spirit which I think has brought so many parts of our country together.
"They should be -- and I think they are -- incredibly proud of what they've done. For some of them to be abused is unforgivable really. I know a lot of that has come from abroad.
"People who track those things have been able to explain that. But not all of it. And it is just not what we stand for. I think we have been a beacon of light in bringing people together, in people being able to relate to the national team and the national team stands for everybody.
"That togetherness has to continue and we've shown the power our country has when it does come together and has that energy and positivity together. We felt that from the fans and I'm incredibly proud of the players.
"The game, we needed to win in the 120 minutes. We were a little bit short in that period, then it is my decision who takes the penalties. It is not a case of players not volunteering or more experienced players backing out. That was my choice, nobody else's.
"My decision to give the guys the penalties they took. I've still got huge belief in them. Those boys have done a brilliant job and we heal together as a team now. I know that 99% of the public will be as well because they will appreciate how well they've [played].
"Bukayo in particular has been an absolute star in this tournament. Incredible maturity, the way he has played. He's brought a smile to so many people's faces. He's become such a hugely popular member of the group. I know he's got everybody's support."
UEFA echoed Southgate's stance with a Twitter post that read: "UEFA strongly condemns the disgusting racist abuse directed at several England players on social media after the EURO final, which has no place in football or society. We stand by the players and the English FA's call for the strongest possible punishments."
FA chief executive Mark Bullingham indicated before the round of 16 stage that Southgate will be offered an extension to his current contract which expires after the 2022 World Cup.
Southgate confirmed his intention to lead the team into Qatar but refused to indicate whether he would stay on beyond that in a post he originally took up in 2016.
"I don't think now is an appropriate time to think about anything," he said when asked about his future.
"We've got of course to qualify for Qatar. But I need some time to go away, watch last night's game again, reflect on the whole tournament. I need a rest. It's an amazing experience but to lead your country in these tournaments takes its toll and I need a break now.
"I said at the time it was great to have that internal support [from the FA]. You hugely value that as a manager but also there is a lot to think through. It is not about finance in any way or commitment.
"I don't want to commit to anything longer than I should, I never want to outstay my welcome. So all of those things need consideration before even thinking about sitting down and talking. But as I sit here today, I would want to be taking the team to Qatar. I feel we've made progress over the four years.
"We've had a fourth place [at the 2018 World Cup], a third place [at the 2019 Nations League] and a second place. It is probably as good as any other team in Europe bar those that have won the tournaments themselves. But for consistency its right up there.
"A lot of things we've done right and we know this team isn't at its peak yet but that doesn't guarantee winning because we know how difficult it is to get back to the stage we got to last night. That's why it's so painful to get so close."
Britain's Prince William also condemned the racist abuse of England's players.
"I am sickened by the racist abuse aimed at England players after last night's match," William, Queen Elizabeth's grandson and president of the English Football Association, said on Twitter.
"It is totally unacceptable that players have to endure this abhorrent behaviour. It must stop now and all those involved should be held accountable."