The 23-year-old has spent each of the last three seasons on loan with AC Milan, but it was recently confirmed that he would be heading back to Bernabeu this summer.
Diaz has also penned a new contract with Carlo Ancelotti's side until the summer of 2027, with the departures of Karim Benzema, Marco Asensio, Eden Hazard and Mariano Diaz opening the door for him to return.
The Spaniard has said that he has "grown and matured at Milan" and is confident that he will be able to have an impact at Real Madrid this time around.
Diaz moved to Los Blancos from Manchester City in January 2019, but he only managed two goals and three assists in 21 appearances for the club before moving to Milan.
"I've grown and matured at Milan over the last three years both on and off the pitch. I want to be successful at Real Madrid, the best team in the world. I'm very happy to be back," he said at his unveiling on Tuesday.
"My first thought was that it was a long day, but it was my day and I was going to have a great time with my family. It's one of the happiest days of my life. Being here is every child's dream and making it come true for the second time is even more of a dream. I want to fight for this badge because it is the biggest in the world.
"I'm lucky to be able to play with both feet which gives me an advantage. My best position is wherever the coach puts me. Out wide you have more balance and in the centre you can connect with the players up top. I'll be happy wherever the coach puts me.
"Competition is there, but that's a good thing. I'm at the best team in the world, with the best players, and that's good both for the coach and for Real Madrid. I'm obviously coming back to succeed, but always through hard work, humility and the constant desire to learn from the fantastic dressing room here, which continues to win things. I have to work hard and listen to the coach."
Meanwhile, Real Madrid president Florentino Perez added: "He's here with us, making this squad even stronger, one of those great players who possesses the talent and quality of those who are destined to triumph in our jersey.
"You've demonstrated the strength and courage to want to prove from afar that your football, your hard work and your ambition to improve every day would bring you back to Real Madrid.
"And now, at just 23 years of age, you return, but with the experience and fortitude of having performed as one of the best in such a difficult league as is Italian football".
"Today you begin a new chapter in your career full of hope and emotions. Our fans will always stand by you along this journey. They will give you everything they can because we know how hard you have worked to get back in this jersey.
"You've learned to live with the enormous pressure at Real Madrid. However, you're also aware that our values will always guide you through the most difficult moments.
"You belong to a club where the word "give up" doesn't exist. Everything is possible with hard work, respect, humility, teamwork and solidarity. That's how we are renowned around the world and that's why we have won 14 European Cups."
Diaz was in strong form for Milan during the 2022-23 campaign, featuring on 45 occasions in all competitions, scoring seven goals and registering seven assists in the process.