Not even remotely. The two had at best a rivalry, but mostly animosity, and mostly, according to people who knew both, it was McQueen more than Newman.
McQueen and Newman both raced, and both raced well. Newman worked hard at racing, and according to Dick Barbour of Barbour Motorsports, who trained both men, Newman was methodical and disciplined, having to work hard to get performance, whereas McQueen was a natural, and just was that good, picking up the finer nuances of driving immediately. Many say that McQueen hated that Newman was every bit as accomplished on track as he, when he felt he was the better driver. However, the two never faced each other on track.
And it wasnt limited to the race track. McQueen was up for the co-star role in Butch and Sundance, but he felt he should have top billing, even though he wasnt as big of a name as Newman at the time. So he turned down the role,
which eventually went to Redford, though they also tried Warren Beatty and Marlon Brando.
The same issues happened with the only film they did appear in,
Towering Inferno. McQueen again felt he should be the star. According to one lifelong friend of Newman, writer A. E. Hotcher, McQueen went so far as to constantly count his lines in scenes vs Newman, then complain when he didnt get as many or more. He recalls Newman calling McQueen a “chicken shit”.