The 24-year-old, a BT Sport Ambassador, completed his world-record €100 million switch to the Spanish club on Sunday evening and signed a six-year contract with Carlo Ancelotti's side before being unveiled at the Santiago Bernabeu on Monday.
Negotiations which lasted throughout the summer saw Levy attempt to hold out for a figure of €115m, while a late delay in the deal was caused by Tottenham wanting to complete a move for Erik Lamela before allowing Bale to leave.
"It's obviously been a whirlwind few days. It was great to be unveiled, it's a day I'll never forget," Bale told BT Sport.
"It was a long time, it was very stressful to say the least. I knew their interest from the start and I was always confident the move would go through. The chairman, Daniel Levy, had to do his business and do Tottenham well - and I kind of understood that - but, at the same time, it was a hard time.
"I had to be patient and try and focus. I believed the president and Jose [Angel Sanchez] would get the deal through and finally it did and it was great to be unveiled."