Is it ever implied anywhere exactly what happened to Doc Hudson between the events of Cars and Cars 2? At I've point one of the characters says something like "doc would've been proud of you". I assume the reason for the character's exclusion is because Paul Newman didn't get signed up.
Paul Newman died in September 2008 and, although the pre-production on Cars 2 was already in full swing, he was either too ill or already passed on before it came time to do the voice acting.
The scene you mention in your question is a touching homage to the legend that was Newman.
In a Time interview with John Lasseter, he spoke about the reasons why Doc's voice wasn't recast;
Cars 2 includes an homage to Paul Newman, whose character, Doc Hudson, was the elder statesman of Radiator Springs. Newman died shortly after making the original Cars, and now the Piston Cup, McQueen’s big race, has been renamed the Hudson Hornet Piston Cup. Did you cast Michael Caine as Finn McMissile to try to fill those elder-statesman shoes?
First of all, Paul Newman and I became very close friends. I loved that man. I went to a number of races with him and shared his love of racing. Every time I went to New York, I would have dinner with him and Joanne. We have lost a few of our voice actors in the past and had to find soundalikes, but I just couldn’t do that because Doc Hudson was Paul Newman.
So, that’s why I say he was Doc Hudson and Doc Hudson was him. So, when we started contemplating the new movie, right away we said: “Let’s just come up with a tasteful way to somehow pay homage to the character and to Paul Newman.” So, we came up with the idea that the big cup that they race for is called The Piston Cup, which became re-named The Hudson Hornet Memorial Piston Cup in honour of Doc Hudson and they turned his doctor’s office into a Doc Hudson Museum. So, there’s just a little moment with Mater and McQueen right at the beginning which was my kind of tribute to Paul Newman in the film.
That being said, I know of no source that confirms what the Doc actually died of in-universe (presumably simply of old age or the car equivalent of organ failure)