The versatile winger was one of three England players to miss in the shoot-out defeat to Italy, along with Marcus Rashford and Jadon Sancho, and revealed he "knew instantly the kind of hate that I was about to receive".
Gianluigi Donnarumma's save from Saka's spot-kick was the decisive moment in the match, which finished 1-1 after extra time, as England fell to a 3-2 penalties defeat at Wembley.
Saka, Rashford and Sancho were all subjected to racist abuse on social media after the game, while a mural of Rashford was defaced in Manchester, prompting a strong reaction from England team-mates, manager Gareth Southgate and the Football Association.
Rashford and Sancho addressed the situation with messages posted on Monday and Wednesday respectively, and 19 year-old Saka delivered his own powerful message.
"I have stayed away from social media for a few days to spend time with my family and reflect on the last few weeks," he wrote. "This message won't do it justice how grateful I am for all the love that I have received, and I feel that I need to thank everyone who has supported me."
He described his England team-mates as "brothers for life" and added: "There are no words to tell you how disappointed I was with the result and my penalty. I really believed we would win this for you. I'm sorry that we couldn't bring it home for you this year, but I promise you that we will give everything we've got to make sure this generation knows how it feels to win.
"My reaction post match said it all, I was hurting so much and I felt like I'd let you all and my England family down, but I can promise you this... I will not let that moment or the negativity that I've received this week break me."
The Arsenal youngster called out the likes of Instagram, Twitter and Facebook, telling them to do more to tackle problem users.
"For those who have campaigned on my behalf and sent me heartfelt letters, wished me and my family well - I'm so thankful," Saka said.
"This is what football should be about. Passion, people of all races, genders, religions and backgrounds coming together with one shared joy of the rollercoaster of football.
"To the social media platforms @instagram @twitter @facebook I don't want any child or adult to have to receive the hateful and hurtful messages that me Marcus and Jadon have received this week.
"I knew instantly the kind of hate that I was about to receive and that is a sad reality that your powerful platforms are not doing enough to stop these messages.
"There is no place for racism or hate of any kind in football or in any area of society. To the majority of people coming together to call out the people sending these messages, by taking action and reporting these comments to the police and by driving out the hate by being kind to one another, we will win. Love always wins."