The duo has dominated the sport for over a decade, winning a combined 79 trophies and being the only two players to have scored over 800 goals each.
"I don't see things like that, the rivalry is gone," Ronaldo said. "It was good, spectators liked it.
"Those who like Cristiano Ronaldo don't have to hate Messi and vice versa. We've done well, we have changed the history of football. We are respected all over the world, that's the most important thing.
"He's followed his path and I have followed mine, regardless of playing outside of Europe. From what I've seen, he's been doing well and so have I. The legacy lives on, but I don't see the rivalry like that. We shared the stage many times, it was 15 years. I'm not saying we're friends, I've never had dinner with him, but we're professional colleagues and we respect each other."
Ronaldo, meanwhile, won his first title with Saudi Pro League side Al-Nassr, the club he joined in January from Manchester United, when he lifted the Arab Club Champions Cup last month.
"It's a historic landmark," Ronaldo, 38, said. "For me it was a source of pride the numbers I achieved, which I didn't think I could reach.
"But I want more. While playing the game, the bar has to be high, thinking big."
The five-time Ballon d'Or winner is also proud to have become the pioneer for other star players who have joined the Saudi Pro League.
At the start of the summer, Karim Benzema and N'Golo Kanté joined Saudi Pro League defending champions Al Ittihad, Senegal goalkeeper Édouard Mendy signed for Al Ahli, and former Chelsea star Kalidou Koulibaly joined Al Hilal.
Moreover, Marcelo Brozovic moved to Ronaldo's Al-Nassr, Roberto Firmino completed a free transfer move to Al Ahli after leaving Liverpool, and Brazil star Neymar signed for Al Hilal following a €90m transfer from PSG.
"I knew this [transfer market] was going to happen. I said it six months ago and everyone thought I was the crazy one," Ronaldo said. "But, after all, the crazy guy is not that crazy, and it turns out to be normal to play in the Arab league.
"For me, it was a great privilege to change the culture of a country in terms of football and to have great stars go to Saudi Arabia.
"I was the pioneer, and I am proud of it. What I want most is to continue to always evolve, so that [the league] is top notch."