The 24-year-old attacking midfielder was introduced by president Florentino Perez amid enthusiastic chants from the supporters after earlier passing a medical test.
"It is a great dream for me to play for Real Madrid," Bale said in Spanish before thanking the fans for a "great welcome."
"I want to help to bring the team success," Bale said before ending his speech with "Hala Madrid," the same club chant that David Beckham used at his presentation 10 years ago.
A person familiar with the deal said on Sunday that the fee for the Wales international was a world-record $132 million, making the midfielder the world's most expensive transfer. The person spoke on condition of anonymity because financial details are not being disclosed.
By pairing Bale alongside Cristiano Ronaldo, new Madrid coach Carlo Ancelotti will have one of the most formidable attacking combinations in football and one intended to compete with that of Spanish champion Barcelona, which features Lionel Messi and new signing Neymar.
Bale, whose move was the culmination of negotiations that spanned much of the summer, acknowledged his signing had involved a complex process that had made him feel uneasy at times.
"It's been difficult, but I'm here now," he said. "It's been a very nervous situation, but it's a dream come true."
Bale said that the deal had been done correctly, with Real Madrid approaching Tottenham first, before talking to him.
He said he wasn't told until late Sunday that an agreement had finally been reached — shortly before it was announced to the rest of the world.
Standing in front of a large photograph of Bale when he was a little boy, wearing a Madrid shirt, Perez praised the Welsh player's skill.
"We are before an exceptional footballer," said Perez, pointing out that Bale had been chosen as the best player in the Premier League last season. "He is a young player with very great qualities."
Perez said the player had shown "determination and solidity" throughout the negotiations, something he said had helped in completing the transfer process successfully.
"I thank Tottenham for giving us the opportunity to count on this player," Perez said.
Bale acknowledged the transfer fee had been surprisingly large, but said that aspect of the deal was strictly between Tottenham and Madrid and had not included him at any stage.
"I would have come here for a penny," he said. "I just wanted to be here and in a white shirt."
Despite staying away from Tottenham during the final days of negotiations, the player said he had been working on his fitness and was hoping to play for Wales in its two upcoming World Cup qualifiers. He will travel to Skopje to play Macedonia on Friday, followed by a match in Cardiff against Serbia four days later.
Bale insisted that the hefty price tag didn't put any added pressure on him, but added that he already expects a lot from himself and looks forward to pitting his skills against the competition in the league.
"I've always followed Spanish football, it's an amazing league. I had the (Madrid) shirt when I was nine years old," he said.
He said he had already spoken briefly with Ancelotti and was not concerned about where the coach chooses to play him.
"I enjoy being in all the positions I've played in so far, and I don't know if I've got a favourite position anymore," he said. "I'll be happy to play wherever he places me."
Bale, perhaps mindful that his arrival has the potential to unsettle Real's established superstar, said Ronaldo was the best player in the world and that his desire to play alongside the Portuguese international had been "a massive factor" in wanting to join Madrid.
"I want to learn off him," Bale said.
With Madrid, Bale will also be playing in the Champions League again — a competition that Tottenham only reached once during his stint there. He will be wearing the same No. 11 shirt that was worn by Francisco "Paco" Gento, who helped Madrid win six European Cups in the 1950s and '60s.
Bale said one of his favourite memories of Madrid was Zinedine Zidane scoring the winning goal in its 2-1 Champions League final win against Bayern Leverkusen in 2002.
"Tottenham wasn't easy to leave," he said, but "my heart was always set on Real Madrid."
Rob Harris in London contributed to this report.