I find the idea of anyone considering the Doctor a Gary Stu absolutely hilarious. What it shows is that these days no one has any idea what the term means, and that it (and Mary Sue) have come to be used to describe any character the user of the term dislikes and happens to be somewhat competent at something. Honestly, calling the Doctor a Gary Stu is nearly as stupid as calling Geralt of Rivia one (and I've heard that bandied around a few times, and still find it utterly ridiculous).
Look at it this way. Mary Sue/Gary Stus are meant to be author-insert characters, someone unbelievably awesome, loved by everyone, usually something the author would want to be. Who the hell would want to be the Doctor? Yes, he has the whole of time and space to play with. Yes, he's essentially immortal. Yes, he's damned intelligent. But honestly, wanting those things doesn't make you want to be the Doctor. They make you want to be a Time Lord with a TARDIS (or possibly Captain Jack Harkness, but that's another matter

).
The Doctor, on the other hand, lives a thoroughly unpleasant life. People hate him. Others fear him. His home is destroyed, his people wiped out, the race they sacrificed themselves to kill is still out there, and the Doctor has thoroughly failed in his original attempt to travel the universe at whim. Admittedly, David Tennant's character isn't all that interesting, but that doesn't make him a Gary Stu. That makes him a bland, badly written character. There's a difference between that and an author-insert. I still liked the David Tennant episodes, but that was more due to the actor than the writing (although Season 3 one of the best episodes ever, Blink).
On the subject of whether you should keep watching though, I say go for it. Don't worry, Matt Smith's a much more enjoyable character to watch, and the themes running through the fifth and sixth seasons are more interesting than anything done under RTD. Not to mention that they finally get away from saving the universe every episode and, mostly, spend their time dealing with smaller scale, more personal issues (admittedly, the universe tries to end on them a few times, but no where near as often as in series 2-4).
Edit: Ninja'd on the Blink point.