Rashford, Sancho and Saka all missed spot-kicks in England' s heartbreaking 3-2 shootout defeat to the Italians.
Saka, who missed the decisive penalty in the final, was subjected to racist abuse on Instagram just minutes after the shootout.
Graffiti was scrawled across a mural of Rashford in Manchester less than an hour after the Three Lions' final loss.
England's players have taken a knee before games at UEFA Euro 2020 to highlight the fight against racial inequality.
The Metropolitan Police have opened an investigation into the abuse.
Soutgate said: "I'm not totally across everything but my first thoughts this morning are with the boys that have done so well for us.
"The players have had such a great togetherness and spirit and brought our country together,
"For some of them to be abused is unforgivable. It's just not what we stand for.
"We have been a beacon of light to bring people together and the national team stands for everybody.
"We felt the energy and positivity from the fans and I'm incredibly proud of that.
"I think we’ve 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 must continue. We’ve shown the power the country has when we do come together.
"We heal together as a team now. We’re here for them, and I know 99 per cent of the public will be as well.
"Bukayo in particular has been an absolute star in this tournament. He’s become such a hugely popular member of the group, and I know he’s got everybody’s support."