I don't think there is any software that does what you say exactly in the way that you want it. But if you can relax your requirements a little there is something that may work for you.
Instead of just looking at a single image and giving it a score, let's say you have a golden image, one that you have inspected in every way and you consider it the best it can be for its class.
Now you can compare other images of the same subject (with same focal length, same exposure, etc.) and get a score for how close the new images are to the golden one. For example, you could use this technique to objectively measure how much the JPEG compression degraded an image, using the equivalent raw image as your golden image. (note that I said objective, this algorithm does not take into account subjective aspects of a comparison, so it isn't sometimes an indication of perceived quality).
The algorithm is called Peak signal-to-noise ratio or PSNR in short. There are several open source implementations of this algorithm. OpenCV and FFmpeg have them, this question at stackoverflow.com lists a few more.