Aosta is a wonderful place to spend a night or two. The method for completing the trip from Aguille du Midi is to catch the gondola to Helbronner and then down into Courmayeur, rather than into Palud, which was a bit of a distance into Courmayeur in the past. From there, you can get the bus into Aosta, which takes about an hour. Here's the problem, timewise. Your trip from Geneva to Chamonix takes 1 1/2 to 2 hours. Then you get to wait in line for the cable car to Aguille du Midi. Depending upon when you leave Geneva and when you arrive in Chamonix, this could be a lengthy wait, especially in summertime. I took the first cable car at around 8:00 or so in the morning, and there was a decent line at 7:30 when I arrived at the station. Figure an hour wait at the minimum, could be more. The trip up the mountain to the Aguille is about 20 minutes, but then figure another hour or so to Helbronner. I don't know how long the Helbronner cable car takes, but let's estimate 30 minutes down, assuming you catch it just right. That's roughly 5 hours so far from Geneva to Courmayeur, assuming you hit everything perfectly. Add another hour into Aosta, and you have a 6 hour day under your belt without seeing Aosta at all. Figure another 2 hrs or so back to Geneva, and it's a full day.
I would do the trip with a small bag and spend the night, returning the next day. The cable car ride was amazing, but it doesn't go to Aosta.