Derleth covered it pretty well, but I’ll see what I can come up with off the top of my head. Ok, the Shuttle is 230,000 lbs at landing, so that’s the approx. dry weight of the Orbiter (what we call the part that most people think of as the Shuttle). It would actually be more with the second stage to get it from floating to space, but that gives you an idea of the numbers.
Now think of how big a blimp is. That’s neutrally buoyant - how much bigger would it have to be to lift a quarter million pounds? And how are you planning to fill all of that in a controlled fashion? How will you attach something that big to the spacecraft?
As you ascend the atmosphere gets thinner, which means that the pressure outside of the balloon will drop, making the balloon want to expand. I’m not sure why, but this seems significant.
Now, in order to get the orbital slot you want, you have to set off the second stage at a specific time, pointed in a specific direction, while in a specific location. What happens if the wind is blowing in the wrong direction during the half-hour you have to launch? Of course, you’ll have to have some sort of navigation system, telemetry, etc. which will only add to the amount of lift you need. Also, the balloon may float, but it still has inertia. It doesn’t exactly turn on a dime.
Finally, as Derleth pointed out, having expended launch vehicle fall on the populace is generally frowned upon. Therefore, most countries launch over unoccupied water or land. The falling balloon would be very difficult, if not impossible to control or even predict, especially after having a rocket go off right next to it.
In summation, probably possible, probably not the best solution.
Also, a little detail - the rocket is called a launch vehicle and that’s what’s filled with fuel. The spacecraft is the thing at the top of the rocket and only has a little bit of fuel comparatively.
On preview, thanks to MonkeyMensch for bringing up the whole altitude vs. horizontal velocity thing. No way was I up to that at this time of the night.